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Functional design and scaling of the jumping mechanism of the African desert locust Katz, Stephen L. 1993

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FUNCTIONAL DESIGN AND SCALING OF THE JUMPING MECHANISM OF THEAFRICAN DESERT LOCUST (Schistocerca gregaria).byStephen Lawrence KatzB.A. Occidental College, 1986.A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THEDEGREE OF DOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of ZoologyWe accept this thesis as conforming to the required standardThe University of British ColumbiaSeptember 1992.© Stephen L. Katz, 1992.In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of The University of British ColumbiaVancouver, CanadaDate nv- CZ, (.992DE-6 (2/88)ABSTRACTCurrent models for scaling of skeletal morphology were examined to testtheir applicability to the ontogenetic growth of an exoskeletal animal, theAfrican Desert Locust (Schistocerca gregaria). It was found that the tibial legsegments of both the mesothoracic (ie. non-jumping) and the metathoracic(jumping) legs scaled in a manner that produced relatively longer, more slenderskeletal elements as the animal grew. Metathoracic tibial length scaled to tibialdiameter raised to the power 1.21. This result deviates both from isometric (ie.geometric similarity) and distortive (constant stress, elastic similarity) allometricmodels.The mechanical properties of the metathoracic tibiae were measuredusing a dynamic, 3-point bending technique. The flexural stiffness ofmetathoracic tibiae scaled to body mass raised to the power of 1.53. This wasintermediate to the predictions made by constant stress and elastic similaritymodels. Thus, the mechanical properties scaled in a manner similar topredictions of mechanical scaling expectations in spite of the morphological,developmental programme.The uncoupling between morphological scaling and structural,mechanical properties is accomplished by scaling the tensile modulus of thecuticle. This strategy of altering the properties of the building material is distinctfrom strategies employed by vertebrate animals.Calculations indicate that the energy stored in the substantial deflectionof the adult, metathoracic tibiae during the jump may be as high as 10% of thetotal kinetic energy of the jump.Using the models that generated relationships between morphology and-II-body size proposed by McMahon (1973, 1984) and the relationships betweenmorphology and performance described by Hill (1950), predictions of howjumping performance measures may change as a function of body mass weretested. Performance was quantified using a high sensitivity, three dimensionalforceplate. Performance parameters quantified included the force,acceleration, take-off velocity, kinetic energy and power output. With theexception of power output, each measure of performance scaled to body massin a manner consistent with the predictions of the elastic similarity model. Poweroutput scaled to body mass in a manner consistent with the predictions of theconstant stress similarity model. The elastic similarity model is approximated bythe performance of the locust in spite of the morphological design that deviatesfrom that model's predictions.These results indicate that the jump has separate functions in the flightless,juvenile instars and in the flying adult stage of the life history. Juvenile locustsproduce take-off velocities of between .9 and 1.2 m/s that are relatively sizeindependent. The take-off velocity in juveniles produces a distance of ballistictravel that averages between 20 and 30 cm. In adults, the take-off velocity isalso relatively size independent at a level approximately twice as high as injuveniles (ie 2.5 m/s). The data suggest that in juveniles, the jump is designed toachieve a characteristic distance travelled, and in adults the jump is designedto achieve a minimum velocity necessary to fly.Three rationales for the observed morphological programme are offered.The design may be a manifestation of a developmental constraint, it may be aresponse to scaling to force rather than explicitly to body mass, or it may be adesign that takes advantage of the inherent deformability of long, slenderbeams. Thus, it may be that the tibiae, which have been treated as rigid levers,are in fact flexible springs.TABLE OF CONTENTSAbstract^ iiTable of Contents^ ivList of Figures viiList of Tables^ xiAcknowledgement^ xiiPreface^ xivChapter One^GENERAL INTRODUCTION^ 1Why is Size Important? 2Scaling Models in Animal Design^4Geometric Similarity 6Constant Stress Similarity^ 8Elastic Similarity^ 10Do Animals Follow Scaling Models?^14Locust Jumping^ 17Chapter Two^^THE UNCOUPLING OF MORPHOLOGICAL AND MECHANICALSCALINGINTRODUCTION^ 21METHODS And MATERIALSAnimal Husbandry^ 22Morphological Measurements^23Dynamic Mechanical Measurements^26Statistics^ 31RESULTS^ 31Morphology^ 34Mechanical Measurements^40-iv-DISCUSSION^ 46Exo- vs. Endoskeletal Design^49Scaling in Cursorial & Jumping Insects^53Ontogenetic vs. Phylogenetic Scaling^55Design in Jumping Tibiae^57Chapter Three SCALING MODULUS AS A DEGREE OF FREEDOMINTRODUCTION^ 62METHODS And MATERIALS^ 67RESULTSWithin Instar Changes in I^69Scaling^ 72Orientation of Neutral Axis^73DISCUSSIONWithin Instar Changes in I^84Scaling^ 85Orientation of Neutral Axis^87Chapter Four HOW HARD DO LOCUSTS JUMP?INTRODUCTION^ 90METHODS And MATERIALSAnimal Husbandry^ 93The Force Plate 96Jumping Arena^ 97Data Analysis 98Statistics^ 98RESULTS^ 100-v-DISCUSSIONScaling of Locomotor Performance^123Acceleration and Design^133The Ontogenetic Role of Jump Performance139Chapter Five^GENERAL DISCUSSION^ 146The Manufacturing Issue^147Force vs. Body Mass Scaling 148Spindly Levers as Design Strategies^152Degrees of Freedom in Design 166Chapter Six^CONCLUSIONS^ 167Literature Cited 169Figure Figure 2.1Figure 2.2Figure 2.3Figure 2.4Figure 2.5Figure 2.6Figure 2.7Figure 2.8Figure 2.9Figure 2.10Figure 2.11LIST OF FIGURESTitle^ Pace Diagram showing the anatomical landmarks used in 25measuring length and diameter of tibiae.Diagram of the mechanical testing apparatus designed to 28deliver a dynamic three-point bend.The relationship of body mass to the age of the locust.^32Plot of tibial lengths and diameters for mesothoracic and 35metathoracic legs with increasing age.The relationship between the log of tibial length and the 36log of body mass for mesothoracic and metathoracic legs.The relationship between the log of tibial diameters and 37the log of body mass for mesothoracic and metathoraciclegs.The relationship between the log of tibial length and the 39log of tibial diameters for mesothoracic and metathoraciclegs.Mechanical test data showing log of flexural stiffness as a 41function of the log of frequency of the imposeddeformation.Changes in the log of flexural stiffness with age.^42The relationship between the log of flexural stiffness and the 44log of body mass.The time course of changes in resilience of the 45metathoracic tibia with age.-vii-Figure 2.12 Theoretical model of optimal internal to external diameter 52ratio (k) for hollow skeletal structures with various luminalcontent to wall material specific gravity (Sg).Figure 3.1 A diagram of a beam of irregular cros-section to show the 65calculation of the second moment of area (I).Figure 3.2 The relationship between the log of flexural stiffness and 70age within the fifth instar.Figure 3.3 The relationship between the second moment of area (I) 71for the various cuticular components and age within thefifth instar.Figure 3.4 The relationship between the log of second moment of 74area (I) for the various cuticular components and the logof body mass.Figure 3.5 The relationship between the log of tensile modulus (E') 75and the log of body mass.Figure 3.6 Example of cross-section from a fifth instar locust metatibia 77and binary images of the cuticular components.Figure 3.7 Plot of the relationship between I and the orientation of the 79neutral axis for a sample of six legs and their averagevalue.Figure 3.8 Comparison of the effect of orientation of the neutral axis 80on I for an example leg and an elliptical cross-section ofsimilar aspect ratio.Figure 3.9 Example cross-section from the same leg as shown in figure 823.6 demonstrating the location of spurs.Figure 3.10 Plot of the relationship between I and the orientation of the 83neutral axis for an average leg with spurs projecting fromalternate sides.Figure 4.1 Diagram of the force plate used in this study.^95Figure 4.2 Sample data from a 0.5007 gram fourth instar showing time 99course of force, velocity, movement and power productionFigure 4.3 The relationships between body mass, measured ground 103reaction force and peak acceleration, and age of thelocust.Figure 4.4 The relationships between the log of ground reaction 107force, peak acceleration and jump impulse duration andthe log of body mass.Figure 4.5 The relationships between velocity produced in the jump 110and age and between the log of velocity and the log ofbody mass.Figure 4.6 The relationships between kinetic energy produced in the 113jump and age and between the log of kinetic energy andthe log of body mass.Figure 4.7 The relationships between average and peak power 116produced in the jump and age and between the log ofaverage and peak power and the log of body mass.Figure 4.8 The relationships between average and peak mass specific 119power produced in the jump and age and between thelog of average and peak mass specific power and the logof body mass.Figure 4.9Figure 4.10Figure 4.11Figure 4.12Figure 4.13Figure 5.1Figure 5.2Figure 5.3Figure 5.4Figure 5.5Figure 5.6The relationship between the log of the movement of the 121locusts' centres of gravity during the jump impulse and thelog of body mass.The relationship between the trajectory angle of the jump 122and the age of the locust.The relationship between the estimates of the distance 124covered by the locusts as ballistic objects after they leavethe ground and their ages.The relationship between the log of peak acceleration 135produced in jumping and the log of body mass for aselection of invertebrate and vertebrate hoppers.Increment of additional velocity required to produce thrust 144for actuator discs moving at various velocities.Diagrammatic expression of the functional role of the tibiae 154as an energy 'time machine'.Diagram of the mechanical model used to test the idea 157that compliant levers have a functional role in delaying thepoint of peak force.Plot of the displacements of the mass in figure 5.2 through 159time for rigid and compliant levers.Plot of the velocity of the mass in figure 5.2 through time for 160rigid and compliant levers.Plot of the accelerations of the mass in figure 5.2 through 162time for rigid and compliant levers.The relationship between the normalized position in time of 165the peak of acceleration produced in the jump (t lc,/ttotal)and the magnitude of the peak acceleration (amax).-x-LIST OF TABLESTable 4.1 Comparison of observed and predicted scaling exponents. 126ACKNOWLEDGEMENTSSix years in one place and I'm supposed to be able to appropriatelyexpress my thanks to all of the people that contributed to my work here in acouple of pages....yah right.Of the hominids that have contributed to my work the single mostimportant individual is my supervisor John Gosline, who helped pull my graduatecareer off the rocks when it had floundered in rough weather. John has givenme the freedom to do whatever I wanted, and I now appreciate that for thetwo-edged weopon that it is. John deserves a lot of credit for helping meorganize much of the material in this thesis, and for first asking me 'How doexoskeletal animals scale?' (or words to that effect)I would also like to thank my committee members who have at varioustimes been Robert Blake, Bill Milsom, Peter Hochachka, Don McPhail, DaveJones, and Dave Randall. I would especially like to thank Bill Milsom for hiseditorial conscience that was put through its paces on most of the pages of thisthesis.I would also particularly like to acknowledge the contribution of RobertBlake. Of all the people in the "Big House', Bob always took me and my ideasseriously and always read my writing with care and speed. Anyone familiar withBob will also recognize his important contribution to the discussion of actuatordisks in chapter four.There are several of my lab mates and friends that deserve credit forlightening the load in many ways. Among them I would like to thank PaulGuerette, Margo Lillie, Dottie Pabst, Calvin Rosskelly, Ignacio Valenzuela, ClaudiaKasserra and her Chris's, Phil Davies, Troy Day, Laura Nagel and Joel Sawada.Of these Margo Lillie and Paul Guerette deserve special mention. It is Margothat makes the Gosline lab go when John is on his extended trips to thedepartment of zoology at UBC, and it is Margo who I ask to critique my workwhenever I feel the need to be dressed down by someone with a truly largecortex. Junior needs to be thanked simply for being so damned def, but alsofor reminding me not to lose my humanity in the process of doing my work (andfor pointing out what a windbag D'Arcy Thompson was).I would like to thank the people in the huts for reminding me that one cando science and still be a human being. I had been in the Big House so long Ihad forgotten. Sam Gopaul also deserves a big thank you for always havinglocusts for me in large numbers.I would also like to thank Gillian and Izzie Muir Ozzie is the most importantnon-Hominid in my stay at UBC) for listening to my ramblings and being myfriends. Working with Gillian has been rejuvinating intelectual experience;collaborating with a person of Gillian's formidable intelect was stimulating andfun. I would like to thank Izzie for the conversations we've had, and for alwaysagreeing with me.The two people who have played far and away the largest roles in my lifein the recent past, academic and otherwise, have been Mike Hedrick andPatricia Kruk.Mike sucked me into the laugh-a-minute world of fish breathing and itresulted in what I think was a very productive collaboration. Our collaborationprovided me with a much needed vacation from locust legs and an opportunityto work with my best friend. Many people have expressed surprise at the baudrate of Mike's and my communications, and they have commented on theparallel lives we have led. They don't know the half of it.Although she may not appreciate it now, Patricia Kruk was my best friendand confidant for the five years we were together. She showed by examplehow hard a graduate student could work, how a high standard could beproduced, and how to be serious about science. Far more significant was hercontinuous and unfaultering support for me and my brain. I am sure that I wouldnever be able to thank her appropriately, even if things had worked outdifferently.The two people who have contributed over my entire life to what I amnow are my parents Dr.'s David and Carol Katz, and for this there could neverbe an appropriate thank you. I feel that whatever element of perfectionism,curiosity, confidence, dedication to logic, honesty and unwillingness to settle forother people's expectations that are a part of me are expressions of theirinfluence. I will also never be able to thank them enough for always beingsupportive of me (financially and in all other ways) whenever I got in a jam.PREFACELarge parts of chapter two of this thesis has been published in the Journalof Experimental Biology. The full reference data are:Katz S. L. & Gosline, J. M., (1992) Ontoaenetic scaling and mechanical behaviourof the tibiae of the african desert locust (Schistocerca areaaria). J. Exp. Biol.168:125-150.The contribution to this work made by S. L. Katz consisted of the designand construction of the testing apparatus, design and execution of the testingprotocol, analyzing the data, and writing the text of the first draft of themanuscript.To establish concurrence on this statement both authors have signedbelow.The following is a release of copyright from the assignee to the authorallowing the inclusion of the published material in this thesis.CHAPTER 1.GENERAL INTRODUCTION".. But the physicist proclaims aloud that the physical phenomena whichmeet us by the way have their forms not less beautiful and scarce less varied thanthose which move us to admiration among living things. The waves of the sea, thelittle ripples on the shore, the sweeping curve of the sandy bay between theheadlands, the outline of the hills, the shape of the clouds, all these are so manyriddles of form, so many problems of morphology, and all of them the physicistcan more or less easily read and adequately solve: solving them by reference totheir antecedent phenomena, in the material system of mechanical forces towhich they belong, and to which we interprtt them as being due. ... Nor is itotherwise with the material forms of living things. "D'Arcy Thompson, On Growth and Form (1961)This thesis is a study of mechanical design in animals. The word 'design'is so important an idea to this thesis, and yet its use is fraught with teleologicalperil. Design means different things in different contexts. While it can mean theprocess of planning and constructing a device, it also may simply refer to thedevice itself. Evolution is a process of generating (ie. designing) biologicalstructures, but each of these structures represents a design that we canevaluate with respect to the relationship between its structure and its function.Structures or mechanisms can be 'good' mechanical designs, which functionwith ease and efficiency, or they can be bad' designs, which are ill equippedto meet the demands placed upon them. Engineering provides tools that wecan use to evaluate the quality of a given design. 'Mechanical design' for mypurposes is the relationship between the structure and the function of a specificbiological mechanism whose mechanical behaviour can be evaluated withengineering mechanics.Does the use of 'design' imply the desire to find optimality? Notnecessarily. Were we to design a structure for use by people we might sit downand determine with high precision the loads that the device would have toendure and then build a device that meets some criteria for safety andefficiency, such as minimum weight for unit strength. While natural selection isfairly adept at weeding out 'bad' designs, it probably does not design 'good'ones by planning it out as might a human engineer. Selection works on thematerial that it has at hand. As such, optimality will exist only within theconstraints of the available design options for a particular biologicalmechanism, and may not represent any globally recognised optimum. For agiven biological device, departures from our pre-notion of optimality oftenhighlights important compromises in the manufacture and use of biologicalstructures. Thus, any particular structure (ie. design) may not be the optimalsolution to any narrowly defined objective.This thesis is an analysis of the functional morphology of the legs of theAfrican desert locust and how leg design responds to changes in body size. Ina study of the functional design of a biological mechanism it is fundamental toanalyze the relationship between the mechanism and the physical environmentwith which it interacts. There must also be recognition that the physicalenvironment is not independent of scale.Why is Size Important? Why is it important to study the effect of body size ondesign? The reason is that physical laws, and the way in which those principlesinteract, are scale dependant. A familiar illustration of this assertion is the wayin which the physics of the fluid environment that a sperm whale encounters (ie.the sea) is so profoundly different than that experienced by a single sperm (ie.semen). The whale's swimming is influenced to a certain extent by the viscouscharacter of the water, but the fluid's properties are so significant for the spermthat it is forced to swim as the whale might through asphalt on a hot day(Purcell, 1977). This difference is a manifestation of the changing relationshipbetween viscosity and inertia that result simply because of the difference inscale between sperm and whale sized objects. As a result, the design ofstructures, both man-made and otherwise, must accommodate the changinginfluence of physical laws with changing size. It would be fruitless to test theperformance of a ten centimetre model boat hull intending to extrapolatethose observations to a 300 metre ocean liner without first considering the wayhydrodynamic forces change over that range of scale. The scale dependantnature of the physical environment is no less significant in influencing biologicaldesigns as those man-made.Implicit in this assertion is the idea that physical law influences design. Thisseems self evident in the case of man-made structures. When the influence ofphysical principles on man made structures are ignored or misunderstood planescrash, and boats and floating bridges sink. As Thompson noted ninety yearsago, this is no less true in biological structures. Each analysis of functionalmorphology is an evaluation of the quality of a given design in the context ofthe properties of the structural and material elements of the design and thedemands placed on a given structure by the physical environment.Studying the scaling of mechanisms, biological and otherwise, can servetwo functions. First, it can provide a check on our understanding of a givendesign. If we truly understand how a biological mechanism works, then wewould be able to predict accurately how that design is altered toaccommodate changes in body size. If our predictions are validated then weprobably do understand how the design works. If not, the study of scaling mayserve its second purpose and reveal the heretofore unrecognized relativesignificance of competing design pressures, or even entirely new designprinciples.In this study I have examined the ontogenetic scaling of the jumpingmechanism of the African Desert Locust (Schistocerca gregaria), an animal withsix discrete life history stages, or instars, and which varies in body mass by twoand a half orders of magnitude. This scaling analysis provides some interestingcontrasts to the scaling relationships observed in vertebrate skeletal designs inseveral respects; the locust has a protein exoskeleton rather than a mineralizedendoskeleton, changes of external dimensions occur in discrete steps ratherthan continuously, and comparisons are made across the ontogeny of hoppinganimals rather than between closely related, adult walkers and runners. Withthese contrasts in mind, I have examined scaling in both the morphology andmechanical behaviour of the jumping mechanism in Schistocerca to see if thereare distinctive design principles employed in the jumping legs of the locust.Scaling models in animal design. The desire to understand the scalingof skeletal design has resulted in several models that attempt to predict howanimal skeletons are modified to respond to the changing demands ofincreasing body size. The models are distinguished by the specific design issuesthat are thought to act as pressures in generating and refining a specific design.Each model predicts how the skeleton should scale if a given mechanicalparameter, such as stress or deformation, is similar across the range of bodysizes. Thus the predictions of each model exist within a specific context, and itis the assumptions and the context of the model that are being tested in ascaling analysis. This point is non-trivial, as I will continually make comparisons-4-between the data on locusts and the various models; each time testing to seeif the design of the locusts' legs are responding to specific design issues implicitin each model. For this reason I will take the time to more fully detail the threemost prominent models in skeletal design: Geometric Similarity, Constant StressSimilarity and Elastic Similarity.This approach is not without controversy. Both the statistics and thescientific approach have been questioned in the study of allometry. Smith(1980), in a paper that attempted to catalogue the dangers of theindescriminant use of allometry, has suggested that the familiar methods ofestimating parameters of power law functions are inappropriate because of theassumptions made in Huxley's (1932) use of allometric models. He suggests thatoften linear regression applied to non-transformed data make just as good adescription of the data, indicating that a power law may not be an appropriatemodel for the biological phenomena being described. Zar (1968) has pointedout that variances are resolved in log-log transformations in a manner thatmakes use of the least squares' regression a dubious technique in estimatingslopes of regression of power functions. He states that an alternative model toleast squares, that accommodates the variance in both the independent andthe dependant variables, is a better choice for allometric models.A more significant issue than statistical description is perhaps theidentification of an alternative hypothesis in the analysis of the scaling models.Gould (1975) has pointed out that geometric similarity (the specifics of which willbe detailed shortly) is treated as an alternative hypothesis in testing allometricmodels. He states that allometric regression is a 'criterion of subtraction',meaning that the regression line provides information on the influence of bodymass independent of any specific adaptation of animals at any specific pointin the range of body masses. Thus, an observed allometry may hidefundamental differences between large animals and small ones that are notstrictly structural design consequences of body size. Smith (1980) echoes this aswell as other criticisms, and adds that even in the case where the statisticaltechnique reveals a characteristic power function relationship, this by itself doesnot indicate a specific functional relationship.I believe that in this study the use of allometric models is appropriatelyperformed for several reasons. In the case of each model that follows we cana priori establish a mechanical basis for the power law relationship. Further, wewill discover that the variances are positively correlated with the means (ie. thevariances are independent of the means after log transformation). For thesereasons, a log-log estimation of the power function model is justified (Sokal andRholf, 1981). In chapter four we will see that there are important deviations inadult locusts' jump performance that are not strictly a function of body size.These data are excluded from the scaling analysis. We can establish, however,that the remaining data do represent functionally equivalent individuals, andare reasonably analyzed with Gould's criterion of subtraction. The criterion offunctional equivalence will be used throughout this thesis as a basis for theappropriateness of allometric analysis, and data that do not meet this criterionwill be excluded from analysis.Geometric Similarity. The Geometric Similarity Model (GSM) is amathematical formalization of the idea of Euclidean similarity or isometry.Euclidean similarity means that geometry is maintained independent of the sizeof the object being considered (Thompson, 1947). Therefore, the relationshipsbetween all of the dimensions of the object are the same independent of scale,and differences in size result from scalar multiplication of all of the object'sdimensions by a single multiplier. A familiar example is a comparison betweensimilar triangles of different size. The angles at the vertices are the same (thedefinition of similarity in trigonometry), and thus if the base of one of thetriangles is twice the length of the other, then all of the linear dimensions aretwice as large. When applied to biological systems the outcome is the same.Simply, large animals look like geometrically similar small ones.Geometric similarity has consequences that can place real limitations onbiological design. A familiar example will illustrate these as well as show howthe scaling predictions are generated. Given two cubes with one having edgesthat are twice the length of the other, it is probably familiar that the surfacearea of the larger cube will be four times that of the smaller and the volume ofthe larger will be eight times the volume of the smaller. ThusArea — Characteristic length (! )2andVolume — (3 .Therefore, if animals are geometrically similar their linear dimensions will scalewith body mass raised to the 0.333 power, while their surface areas will scale tobody mass raised to the 0.667 power.One of the significant consequences of geometrically similar increases inthe size of skeletal support structures is that the demands of mechanical loadingincrease faster than the ability of the skeleton to accommodate those loads.More specifically, inertial loading is a function of mass (that is volume), a thirdpower function of characteristic dimension, while the cross-sectional area ofmaterial that carries that load is only a second power of characteristic length.In this way geometric similarity is distinct from the other scaling models in that-7-it is not a predictive model based on the control or similarity of any mechanicalparameter, but is rather the formalization of the consequences of conservinggeometry independent of size.A solution to the increasing mechanical demands placed ongeometrically similar structures is to produce distortions in the dimensions of theskeletal design to increase the amount of load-bearing material, and in sodoing accommodate the increasing inertial loads. This was recognised as adesign strategy by Galileo as early as 1638 (Thompson, 1917; Schmidt-Nielsen,1984). Indeed, it was over one hundred years ago that an explicit descriptionwas made of how the diameters of trees must increase with the 3/2 power oftheir length in order to produce similar mechanical behaviour (Greenhill, 1881).More recently two models have been proposed that predict how characteristicdimensions of biological structures should deviate from geometric similarity toproduce structures that maintain either similar normalized load (ie. Stress)(McMahon, 1984), or similar normalized deformations (ie. Strain) (McMahon,1973) independent of scale. Because of the significance of these models to thematerial that will follow I will briefly detail each. In presenting a detaileddescription of these models in his own words, McMahon (1984) demonstrateshow the predictions of the models can be arrived at with a number of differentapproaches. I will only develop each model in a single way that seems mostappropriate for the context of locust leg design.Constant Stress Similarity. As stated above, the loads associated withbody mass increase with increasing size, but stress (a) , or the force normalizedby the area over which that force is distributed, increases even more quickly.Given the scale independence of specific gravity, forces that are associatedwith gravity or inertia, which is to say associated with mass, in geometricallysimilar skeletons scales to the third power of a characteristic length, while areascale to the second power of length. Thus stress must scale to the 3/2 powerof a characteristic length dimension. McMahon (1984) suggested that byappropriately scaling diameter of a cylindrically shaped skeletal memberseparately from the scaling of length, a given design could achieve a constantstress in structures independent of size. In achieving constant stress, largedepartures from geometrically similar morphologies are produced.The specific morphological predictions of the constant stress similaritymodel (CSSM) for the scaling of dimensions of bending beams can begenerated starting with the following relationship from engineering:a = (Ft v)  Eq. 1.1.Iwhere a is the peak stress developed in a beam of length, (and diameter, 2yloaded with a force, F. I is the second moment of area - a parameter thatdescribes both the amount of load-bearing material and how that material isdistributed (Wainwright et al., 1976). The notion of I is central to the subject ofthis thesis and will be developed in more detail in chapters two and three.Suffice to say at this point that the computational formula for I for a cylindricalbeam of radius, r, is:i = ire/ or irc14 Eq. 1.2.4 64Meaning that I — d 4* which has a number of consequences. One of which is wenow have a relationship between all of the parameters that determine the stressin a beam and morphological dimensions. If force is proportional to mass, andtherefore volume, then we can insert these morphological relationships intoequation 1.1 and producea — (d20(/)(d) = d3F = (2^Eq. 1.3.d4^d4 dand if a is a constant (ie. constant stress) thend — /2^Eq. 1.4.So in the CSSM diameter scales to the second power of length. By substitutingthe relationship mass (M) — d2rwe can relate the scaling of both dimensions tobody mass thusly:as 162 = id... = (5d2 11.41 Mt r ISince a is a constantEq.1.5.f oc m0.20^ Eq. 1.6.and likewised . Ma4° .^ Eq. 1.7.This means that to build different sized skeletons of the same building materialthat will experience the same stresses in bending demands that the externaldimension of diameter will have to scale to mass raised to the 0.40 power whilelength will have to scale to mass raised to the 0.20 power. Therefore, the CSSMis distinct from GSM in predicting that as structures become increasingly largethey will become increasingly stout.Elastic Similarity. In 1973 McMahon formalized another alternative togeometric similarity which has been referred to as the elastic similarity model(ESM) (McMahon, 1973; Schmidt-Nielsen, 1984). Elastic similarity predicts- 1 0-morphological scaling relationships based on keeping the deformation per unitlength in a bending beam scale independent, rather than peak stress as wasthe case for CSSM. By trying to establish a similarity based on a different indexof mechanical behaviour, ESM makes a different set of predictions than GSMor CSSM. The deflection per unit length of a cantilever loaded at its end canbe calculated with the following formula:D = Fe^ Eq. 1.8.f^3E1where D is the deflection of a beam of length, 1, loaded at its end with a force,F. E is the elastic modulus of the material out of which the beam is made. E isa property of the building material, and if we assume that little skeletons andlarge skeletons are made from the same material with the same properties, thenthe term 3E is scale independent and falls out of a proportionality based onmorphological dimensions. Elastic similarity is based on D/f being a constantindependent of size and thus we can generate the proportionality:Ff2 . Constant (C 1 )^Eq. 1.9.1which we can relate to morphological dimensions as we did for CSSMC I — FP . (d20((2) I^c14sof3 or d2 oc f3—2dWhich is precisely the result produced by Greenhill (1881) for the design of trees.By once again substituting the proportionality Mass — d 2f, we can make specificscaling relationships for each dimension in terms of mass.M ec d2( so(M)3 . C2 so M3 . d2d2^d6orM3 oc d8 SO d oc M318 or mum.And likewise:1.3 e'c ( M 1 2 or ( cc m114 or M°25°13 ITherefore, if we were to engineer different sized skeletal designs to respond tomass-dependant, bending loads with a constant deflection per unit length ofbeam we would scale the diameter to the 0.375 power of body mass whilescaling lengths to the 0.25 power of body mass. Similarly, if we observedbiological structures scaling in this manner we might infer that selection wasresponding to a design pressure that placed a value on similar normalizeddeflections in beams loaded in bending.It is important to remember that the predictions of elastic similarity areconstructed around the assumption that the beam is acting like a cantileverloaded at its end. For a beam loaded all along its length, the deflection isproportional to the fourth power of length rather than the third (Denny, 1988).Thus the deflection per unit length in this beam is proportional to the third powerof the beam's length rather than the second. For such a beam the predictionsof elastic similarity are identical to those of constant stress. It may turn out thatvertebrate long bones which have muscle origins and insertions along a largepart of their lengths are more appropriately modelled as beams loaded overtheir lengths rather than just at their ends. As we shall see below, locusts' limbbeams are appropriately thought of as loaded at their ends (Heitler, 1974;Bennet-Clark, 1975), and as a result, the predictions of ESM and CSSM may be-12-distinguished.Limb segments that follow the elastic similarity model, and even more sothose that follow constant stress similarity, become relatively more stout as theanimal increases in size. These types of distortions away from geometricsimilarity are characteristic of what McMahon (1984) calls distorting allometries.Indeed Huxley (1932) used the general term 'allometry' to distinguish distortionsaway from geometric similarity, or 'isometry'. Gould (1966) invokes theetymology of the word allometry, which means 'by a different measure' toseparate all those scaling programmes that deviate from isometry, which hepoints out means by the same measure'. In each case, however, the existingscaling strategies predict allometries that produce relatively thicker beams,rather than relatively more slender beams. In chapter two we will discover thatlocust leg growth does not follow any existing allometric model's predictions,but produces increasingly spindly legs.There are also important assumptions integral to all of these models thatmust be acknowledged. The models have assumed a cylindrical geometry, butare used to predict the scaling of distinctly non-cylindrical anatomical structures(McMahon, 1975a, 1975b). In anatomical features with non-circular cross-sections the value of I will still be proportional to the product of four lineardimensions. In elliptical cross-sections for example, I is proportional to one axiscubed times the orthogonal axis. Therefore, as long as relative dimensions ofthe cross-section of a skeletal element are scale independent, that is the shapeof the cross-section is the same independent of size, the predictions of themodels that are based on cylindrical geometry should still be valid. Thefunctional role of violations of this assumption are very important, and will bediscussed in detail in chapter three.-13-Another important assumption in both of these models is that small andlarge skeletal structures are constructed of the same building material.Therefore, the elastic modulus will be the same independent of size. This isprobably an acceptable assumption in the case of vertebrate bones where thematerial properties of the building material are determined largely by themineralized components of the bones (Curry, 1984). In the case of insectcuticle, however, where the mechanical properties are determined by a proteinmaterial whose properties are alterable either by changes in cross-link density(Anderson, 1976), or by hydration state (Vincent, 1980), violations of theassumption of constant E can be significant. It is the exciting possibility that Eis a scaled commodity in structural design that is the subject of chapter three.An additional assumption in making these predictions is that the forcesthat the skeleton is required to support are associated with the body mass, suchas gravitational or inertial loads. In cases where the predominant forces thatthe skeleton must bear are produced by muscular contraction, which arethemselves subject to scale effects, the predictions of the models must beadjusted. The consequences of scaling the skeleton to accommodatemuscular forces rather than body mass are discussed in chapter five.These assumptions, that the building material's mechanical properties andthe strategy of distributing that material will be scale independent, are preciselythe assumptions that lead us to try to make inferences about the mechanicalbehaviour of biological devices from morphological dimensions. We will see inchapter two that in the locust mechanical properties are 'uncoupled' frommorphology, preventing us from making inferences about mechanics based onmorphology, and in chapter three we will learn that this uncoupling results fromviolations of the above assumptions.-14-Do animals follow scaling models?  Before embarking on a detailedanalysis of scaling in locust legs, it is worth asking if any of the models aresupported by observation? In this regard there are' lithe data outside of studiesof vertebrate designs. Bertram and Biewener (1990) have suggested that GSMmay be an appropriate scaling scheme for small animals, where breakage ofthe skeleton due to loading is unlikely. For small animals, having a skeleton thatis sufficiently stiff to function in support and locomotion is probably the primaryutility of the skeleton. They also argue that it is in large animals, where loads arelikely to exceed the breaking strength of geometrically similar skeletons, wherethe morphology must change, or distort, to accommodate those loads. ESMscaling, such a distorting allometry, seems to be observed in the skeletal designof ungulate limb bones (McMahon, 1975), although Bertram and Biewener(1990) suggest that CSSM may more adequately describe the scaling in thelarger members of that data set. It has also been suggested that McMahon'schoice of Bovids was fortuitous in that the order is somewhat singular inexpressing ESM (Alexander et al., 1979, Curry, 1984). In a wide range ofvertebrates with a wide repetoire of locomotor styles Alexander et al. (1979,1981) found a general tendency to following geometric similarity.None of the models has predicted skeletal elements becoming relativelymore slender with increasing size. However, this type of scaling has beenobserved for the scaling of femora and humeri of a wide range of bird species(Prange et al., 1979). It is not clear if this scaling of bird bones represents astrategy per se. Because bird bones may be air filled and thin walled (Curry,1984) they may be able to modulate the mechanical behaviour of their longbones in ways that are different from terrestrial vertebrates and in so doingproduce an unexpected allometry. The functional role of this different skeletal-15-design is discussed in more detail in chapter three. Remembering Gould's(1975) caution however, it is not clear if it is fair to compare hummingbirds,whose humeri are loaded dynamically, at high frequency during hovering withswans, whose humeri are loaded relatively statically in soaring.These relationships have been tested with a variety of interspecificcomparisons between adults of closely related taxa (ie. phylogenetic scaling).One could also analyze the effect of scale in the development of an individualand apply similar relationships (ie. ontogenetic scaling). In his review, Gould(1966) does not distinguish ontogenetic from phylogenetic scaling in the qualityof information that each type of study provides. It is not clear, however, if thedevelopmental programme in a single species produces a series of functionallyequivalent skeletal designs constructed from mechanically equivalent materials(Carrier, 1983, Curry and Pond, 1989). Ontogenetic changes in materialproperties of bone do occur (Curry and Butler, 1975, Carrier, 1983), uncouplingto some extent our ability to infer mechanical behaviour of the skeleton fromits morphology. For example, Carrier's (1983) observations of jack-rabbitssuggest that in ontogeny the rabbit's bone may vary in its material stiffness (ie.modulus) by as much as an order of magnitude. The same data also indicatethat neonates, juveniles and adults may not be functionally equivalent in theirability to locomote. However, if functional equivalence is maintained acrossontogeny, then it should be fair to ask whether the existing models of scalingapply to that ontogenetic sequence.These observations have been made largely on endoskeletal animals,while very little has been said on the scaling of exoskeletal animals. Prange(1977) has shown that leg segments of the cockroach, Periplaneta americana,and the wolf spider, Lycosa lento, scale very closely to GSM over an-16-ontogenetic sequence. This suggests that being exoskeletal does not by itselflimit the applicability of the models.While Bertram and Biewener (1990) have made a case for body size classdetermining the appropriate scaling programme to follow, Bou et al. (1987)have suggested that lifestyle is more important than either body size class orphyletic affinity in determining skeletal scaling. If so, an alternative scaling ofthe skeleton may be demonstrated by an animal that has a locomotormechanism that is distinctly different from that of pedestrians like cockroachesand giraffes. The jump of Schistocerca is a well documented mechanicalbehaviour (Heitler, 1974, Bennet-Clark, 1975, Alexander, 1983) and thereforeprovides a good contextual framework into which an analysis of scaling maybe placed.Locust lumping The jump of the locust has been the subject of intensescrutiny. Bennet-Clark (1975) has shown that the peak power outputs of thelocust jumping muscle is on the order of 450 W/Kg. He has shown for adults,and Gabriel (1985a) has shown for hoppers (ie. juvenile, flightless locusts) thatthe average power outputs of the locust's jump, estimated from jump distance,are all higher than the maximum value for the jumping muscle. Therefore, theyconcluded that a spring mechanism must store the energy for the jumprelatively slowly (ie. at low power outputs) during the inter-jump interval, andrelease the energy quickly during the jump (ie. at high power outputs). Indeed,it has also been demonstrated that the inter-jump interval length is determinedby the time it takes to store a specific amount of energy in the apodemesprings (Steeves & Pearson, 1982). In support of the suggestion that the jump isthe release of stored energy rather than direct muscular action, the jump-17-distance is not temperature dependant while the length of the inter-jumpinterval is in a related grasshopper, Melanoplus bivittatus (Harrison et al., 1991).The implication is that the increased inter-jump interval is the manifestation oflowered muscle power output at low temperatures, while the release of springenergy is unaffected because the material properties of the cuticular springsare not temperature dependant. Heitler (1974) has described the femero-tibialarticulation in great detail. He has shown that the morphology of the joint isdesigned to provide a mechanical catch mechanism that allows the jumpingmuscle to store a large amount of energy without fighting to hold the leg fromextending. The tibial flexor muscle contracts to deform cuticular elements in thejoint, moving the line of action of the extensor muscle tendon system behind theaxis of the joint, producing a 'catch". Thus the muscular contraction that storesenergy in the tendon also helps keep the knee bent.When the locust decides to jump, the tibial flexor muscle relaxes torelease the catch allowing the line of action of the extensor muscle-tendonsystem to apply a torque that extends the knee. The stored energy in theapodeme spring is released, rotating the tibial segment, and extension of theleg pushes on the ground generating a ground reaction force that propels thelocust into the jump.The films of Brown (1963), analyzed by Alexander (1983), indicate thatduring the jump impulse of the adult locust the primary loading regime in themetathoracic tibiae is bending. It seems appropriate, therefore, to investigatethe bending behaviour of limb segments to determine the mechanical scalingof this system. Chapter two will explicitly examine the relationship between thedimensions of the locusts' legs and the mechanical properties of the limb inbending. In chapter two we will see that about 11% of the energy from the-18-spring goes into bending the tibiae, of which about 90% is returned ingenerating ground reaction force. Given the role of bending in the normalloading of the tibiae and the large scale deformation that it undergoes, wemight hypothesize that elastic similarity, as a design strategy, will provide anappropriate model for the legs of locusts. The results of the analysis in chaptertwo suggest that while the tibiae are elastically similar in their mechanicalbehaviour, their external dimensions scale in a manner that is fundamentallydifferent from any of the existing models. This indicates that the relationshipsbetween morphology and mechanical properties that form the basis of thevarious models' predictions are uncoupled in locust tibiae. Thus, to evaluate thetotal mechanical design of a complex, living machine in the context of any onescaling model may be extremely naive.In chapter three this uncoupling is explicitly examined. The mechanicalbehaviour of the locusts' tibiae, or any beam loaded in bending, is a productof the amount, distribution and material properties of the load-bearing material.In chapter three the relative contributions to the leg's bending behaviour thatarise from material properties are separated from those arising from thedistribution of skeletal material. It turns out that material stiffness is scaled in amanner that allows the distinctive morphological scaling that is observed inchapter two.In chapter four the scaling of jump performance of the locust ischaracterized. The results suggest that the jump has a separate functional rolein the juveniles and the adults. In juveniles the results suggest that there is afunctional distance that the hoppers are designed to achieve. The transition towinged adults has demanded a higher performance jump to generate aminimum velocity to begin flight. Within the juvenile instars, however, a-19-consistent expression of the predictions of elastic similarity is observed in thescaling of each of the variables that can be directly correlated withmorphology.In chapter five the significance of the morphological and mechanicaldesign programme expressed by locusts is discussed. Three arguments areoffered to explain the uncoupling that is observed between the morphologicaland the mechanical scaling in chapters two through four. Briefly, the firstargument attempts to describe how the moulting process may form a constrainton the growth process. The second argument suggests that it may be moreappropriate in some cases to scale the dimensions of the skeleton to the forcesproduced by muscles, which themselves scale allometrically, rather thanexplicitly to mass. The third argument will make the case that the relativeslenderness of the legs is functionally important in taking the maximumadvantage of changing mechanical advantage of the legs in the jump.CHAPTER 2.THE UNCOUPLING OF MORPHOLOGICAL AND MECHANICAL SCALING.INTRODUCTION In this chapter I have examined the ontogenetic scaling of themorphology and mechanical behaviour of the legs of the African Desert Locust(Schistocerca gregaria), an animal with six discrete life history stages, or instars,and which varies in body mass by two and a half orders of magnitude. Thisanalysis of the skeleton of the locust provides some interesting contrasts to thescaling relationships observed in vertebrate designs in several respects; thelocust is exoskeletal rather than endoskeletal, changes of external dimensionsoccur in discrete steps rather than continuously, and comparisons are madeacross the ontogeny of one species of hopping insects rather than betweenadults of closely related species of walkers and runners.Regardless of which morphological model is considered, it is assumed thatthe mechanical behaviour of a skeletal member can be inferred frommorphological measurements. In insects where the skeleton is also theintegument and is not mineralized to nearly the same extent as bone, it ispossible that the material properties of the skeleton can be modulated to meetthe mechanical demands of increasing body size in ways that arefundamentally different from vertebrates. The cuticle in various parts of thelocust exoskeleton shows a wide repertoire of mechanical properties (Jensenand Weis-Fogh, 1963, Vincent, 1975), which might suggest such a possibility.Hepburn and Joffe's (1974b) observation that normalized cuticular stiffness ismaintained across instars would argue against the suggestion that material-21-properties of the cuticle are changing. However, insects are a diverse group,and it may be that some of them have developed different strategies forsolving skeletal scaling problems.The films of Brown (1963), analyzed by Alexander (1983), indicate thatduring the jump impulse of the adult locust the primary loading regime in themetathoracic tibiae is bending. I have, therefore, investigated the bendingbehaviour of tibial limb segments to determine the scaling of mechanicalproperties. This study will show that the external skeletal morphology predictsmechanical behaviour that is dramatically different from the observedmechanical behaviour. It will also attempt to.explain how the mechanical resultmay be arrived at in spite of the morphological design programme.METHODS and MATERIALS Animal Husbandry. Animals were sampled daily from a breedingcolony of African Desert Locust (Schistocerca gregaria) maintained at theDepartment of Zoology at the University of British Columbia. The animals werekept at a constant temperature of 27° C, humidity of 56%, and photoperiod of13:11 (L:D), and fed a diet of head lettuce and bran. A sample of fiveindividuals was collected each day beginning on the first day followingemergence from the egg until approximately sexual maturity (ca. 35 days).Each animal contributed both a left and right side meta- and mesothoracictibiae, resulting in four samples from each individual. Replicates wereperformed to increase precision over the first seven, and final 15 days ofsampling. There was no significant heterogeneity between replicates for agiven day, and so all replicates were pooled. As a result, there are differentsample sizes across the time series of morphological and mechanical-22-measurements.Animals were sacrificed by decapitation and weighed immediately to thenearest 0.1 mg. All of the morphological and mechanical measurements werethen performed in air on tibial segments within 10 minutes of removal from theanimal. Control experiments indicated that this time course did not alter themechanical properties significantly due to exposure to air.Morphological measurements. The length and diameter of the tibialsegments of both the mesothoracic, and metathoracic tibiae were measuredeither with a filar micrometer eyepiece (Wild 15XSK) attached to a dissectingmicroscope (Wild M5), or where the tibia length exceeded the length of themicrometer's graticule (the fifth instar and adult tibia lengths), a vernier calliper.Micrometer measurements were made to the nearest 10 gm, and callipermeasurements to the nearest 20 gm. Morphological landmarks were definedto provide ease of location and to indicate the dimensions of uniform limbsegments. That is to say, geometrically complex morphological featuresassociated with the joint articulations were excluded from linear measurement.In the case of the mesothoracic leg, the tibial length was defined as the lengthmeasured on the lateral surface from a point approximately level with thefemero-tibial articulation to the tibio-tarsal articulation. The metathoracic tibiallength was defined from the depression in the posterior surface of the tibia thatoccurs just distal to the knee joint articulation, at the maximal extent ofsclerotized tibial cuticle, down the posterior of the tibia to a point opposite tothe insertion of the first moveable spine at the tibio-tarsal joint. Diameters weredetermined to be the largest diameter of the semi-elliptical cross-section at themid-shaft point. Figure 2.1 shows a graphical representation of the-23-Figure 2.1.Diagram showing the anatomical landmarks used in measuring lengthand diameter of the tibiae. Diameters were measured at the mid length point.L, tibial length; d, tibial diameter.-25-morphological landmarks.Dynamic Mechanical measurements. After morphological measurement,each metathoracic tibia was placed in a mechanical testing frame thatimposed a 3-point load with a dynamic, or time-variant, deformation. Thetheoretical and practical development of this dynamic testing technique havebeen reported previously (DeMont and Gosline, 1988, Lillie and Gosline, 1990).Therefore, only the principles and an outline of modifications made to generate3-point bending will be described. The device itself consists of an actuator thatdelivers the time-variant displacement, monitored by a displacementtransducer, and a force transducer that measures the resultant force developedacross the test piece (ie. a metathoracic tibia) as measured at the two ends ofthe sample. Figure 2.2 is a diagram of the device.The actuator consisted of a length of 0.148' (3.76x1 0"3 m) OD stainless steelhypodermic tubing attached at one end from an electromagnetic vibrator(Model V203, Ling Dynamic Systems, Royston, Hertfordshire, U.K.), and steppeddown at the other end to an 18 gauge hypodermic needle. The end of theneedle was cut off flush and polished smooth. Out of its end protruded alength of 8 lb nylon fishing leader. The nylon loop was adjustable and could beshortened to hold the midshaff of a test piece flush against the end of theactuator by turning a 4-40 machine screw which 'spooled' up the slack nylon.The compliance of the nylon loop introduced less than a 1% overestimate ofactual displacement and was ignored in subsequent calculations.The free ends of the test piece were pulled upwards against the ends ofa window cut in the side wall of a length of 0.148' (3.76x10 3 m) OD stainlesssteel tubing. The tubular holder was attached to a force transducer with a 2-56stainless steel machine screw. The size of the window was scaled to provide a-26-Figure 2.2.Diagram of the mechanical testing apparatus designed to deliver adynamic 3-point bend. a, electromagnetic vibrator; b, inputs for driving noisesignal; c, displacement transducer; d machine screw for tightening the nylonloop that held the tissue; e, stainless steel shaft connecting the vibrator to thetest piece; f, force transducer and attached holder (see inset); g, tubular holder;h, test piece (ie. tibia); i, 18 gauge stainless steel tubing; j, 8 lb. nylon loop; k,cantilever beam of force transducer.ratio of approximately 10:1 of test piece total working length to diameter tomaintain a relatively consistent relationship of bending to shearing moments inthe test pieces. In order to maintain this ratio in legs of different sizes severalholders were made with appropriately sized windows. The ratio was deemeda reasonable compromise between the wish to introduce primarily abending moment with respect to shearing moments, and the difficulties offabricating small holders. The force developed across the test piece wasmeasured with a cantilever-like transducer fabricated out of 0.015' (3.81x10 -4 m)thick stainless steel shim stock. This material provided appropriately smalldeflections (<1% of the imposed displacements). Semiconductor strain gauges(type SR4 SBP3-20-35, BLH Electronics, Canton Mass.) were bonded on bothsurfaces of the cantilever. Semiconductor gauges provided appropriatesensitivity of 0.0473 Newtons/Volt. The resonant frequency of the transducer was1.40 kHz with the smallest holder, 0.95 kHz with the largest, and .65 kHz for theensemble apparatus.As was described previously (DeMont and Gosline, 1988), theelectromagnetic vibrator was driven by the noise generator of a spectrumanalyzer (Model 5820A Cross Channel Spectrum Analyzer, Wavetek RocklandInc., N.J.) which provided a constant power spectrum over the range offrequencies collected (0 - 200 Hz). At each frequency, the spectrum analyzercalculated both the ratio of the amplitudes of the Fourier components of theforce and displacement transducer outputs, and the phase shift (8) betweenthe two signals. Spectra were collected from 0 to 200 Hz, and approximately256 spectra were averaged to produce one spectrum per test piece. Theflexural stiffness (El) of the specimen at each frequency was calculated usingthe following relationship for static 3-point bending:-29-El = F • x3^Eq. 2.1.48 • dwhere x is the length of the test piece (ie. the length of the window in thetubular holder), F is the developed force and d is the deflection at the mid-pointof the beam (rearranged from Gordon, 1978). El is composed of E the elasticstiffness of the beam's material, and I, the beam's second moment of area. Inthese experiments the calibrated amplitudes of the Fourier components of theforce and displacement transducer outputs were employed as F and d inequation 1 to produce a complex flexural stiffness (E .I) (adapted from Ferry,1980).Static tests performed as controls for the dynamic testing showed that thestress-strain curves for tibiae were linear over the range of deformation imposedon the specimen in these experiments.El the storage flexural stiffness--a measure of the energy stored elasticallyper loading cycle, can be found by calculating the in phase component of thecomplex flexural stiffness as follows:El = E •I • Cos 8.^ Eq. 2.2.The energy loss flexural stiffness (E"I) is the out of phase component of thecomplex flexural stiffnessE"I = E . I • sin 8,^ Eq. 2.3.and is a measure of the energy dissipated per loading cycle. The tangent ofthe phase shift (tans = E"I / El) can be used calculate the resilience per 1/2 cycle(R) (in %) of the structure as follows:R = (eta") • 100^ Eq. 2.4.-30-(Wainwright et al., 1976). Explicitly, it is the ratio of the energy recoveredelastically to the energy input to the test piece in each loading cycle. All ofthese calculations were performed on a Digital Equipment Corporation MINC-11/23 computer.Statistics. Except where noted, statistical tests were chosen based oncriteria presented in Sokal and Rohlf (1981). Due to the non-zero varianceassociated with the morphometric variables measured in this study, allregressions were Model II regressions (Sokal and Rohlf, 1981). All Statistical testswere performed with the STATGRAPHICS (STSC, Mass.) statistical softwarepackage.RESULTS Ontogenetic accumulations of body mass followed a characteristicsigmoid curve adequately described by the von Bertalanffy growth function(Pitcher and Hart, 1982). Figure 2.3 shows the daily means of body mass as wellas the fitted curve. The values of sample size for each day, listed at the bottomof figure 2.3, are the same for figures 2.3, 2.4 and 2.7. Body mass ranged from0.0109-103 kg for first day, first instars to 3.541-10 -3 kg in adult, sexually maturefemales. The data show that mass accumulates in a relatively continuousmanner within each instar. Adult locusts continued to accumulate mass for thefirst four to seven days after moulting and then levelled off at their equilibriummass. The assumptions of the von Bertalanffy growth function, that the-31-i 1.,^,■^■^■^■1015151510101010101010101010101010101010101010101020202 0 20202020202020■^■02.92......w1.6ioo1.2rd2i 0.80.9077T-1!11^, 110^20 30^40Age (Days).Figure 2.3.The relationship of body mass to the age of the locust. Data are reportedas means and standard errors of the mean. The regression is fitted to the VonBertalanffy relation with the following form: Mass = 4.027(1-e -1:45 Age)3 (F5=987.8;df=2,33; 12=.962). Numbers listed along the abscissa indicate the sample sizesfor each day, and are the same for figures 2.4a and 2.4b.individual will grow in a logarithmic way up to a point where ingested caloriesare diverted from general somatic growth and to reproductive growth, seemreasonable in this case as growth levelled off at approximately sexual maturity(ca. day 30).All allometric relationships reported here proved to be significant to the0.05 probability level. A significant sexual dimorphism develops in adults wherethe females are approximately 50% more massive than males (ANOVA, F s =21.969, df. = 192, p > 0.05). Analysis of covariance indicated no significanteffect of sex, beyond the effect of mass on any of the morphological ormechanical parameters measured (ANCOVA, F s = 2.451 for mesothoracic tibiaediameter on body mass--the relationship most closely approaching significance,df. = 1,378). Therefore, data from both sexes were pooled in each regression.Morpholoay. Figure 2.4 shows the change in tibial length (Fig. 2.4a.) anddiameter (Fig. 2.4b.) with increasing age. Analysis of variance indicatessignificant heterogeneity between groups, and Newman-Keuls multiple rangetests indicate that each instar is a homogenous, independent group. Thisconfirms that the sample values from the population are reflecting what webelieve to be occurring in individuals, that the external dimensions of leg lengthand diameter are not changing within instars. Analysis of variance of residualsindicated that tibial length and diameter were independent of body masswithin each instar, biasing the overall allometric relationships of length anddiameter against body mass (Draper & Smith, 1981). Therefore, all values for-33-Figure 2.4a.Plot of tibial lengths for mesothoracic (o) and metathoracic (0) legs withincreasing age showing discontinuous growth across instars. Individual pointsrepresent means and 95% confidence intervals to show the similarities within,and differences between instars.Figure 2.4b.Plot of tibial diameters increasing with age. The symbols are the same asin figure 4a.-35-1^4^7^10^13^16^19^22^25Age (Days).28^31^341^4^7^10^13^16^19^22^25^28^31^34Age (Days).EEvoevF202416120 0 0. 0^4>^4> 42)• 4, 4> 4, 0 4.J0434JmcocID^00.000$1:DIDED^00000000000014''It“btt4W+++440$sq§*1.51.20.9C)C0.6F0.3 1:15150rp co r43 $4^4041)4, 4+4,^+00 00m 0 4'0113 t4041 42,4J 431.61.2E0.8Cp0.40-2^-1.5^-1^-0.5^0^0.5Log of Body Mass (grams).Figure 2.5.The relationship of the log of tibial length and the log of body mass formesothoracic (o) and metathoracic (CD legs. Individual points represent meansand standard errors for each variable for each instar. The equation of theregression for the mesothoracic legs is y = 0.868 + 0.356 x X (Fs=850.3; df=1 A;r2=.9963). The equation of the line for the metathoracic legs is y = 1.195 + 0.377x X (Fs=1074.1; df.=1 A; r2=.9953). The dashed lines are the 95% confidence limitof the regression line.1:- -0.1Eea-0.304 ° —0.50.1—0.7—2^—1.5^—1^—0.5^0^0.5Log of Body Mass (grams).Figure 2.6.The relationship of the log of tibial diameter and the log of body mass formesothoracic (o) and metathoracic (0) legs. Individual points represent meansand standard errors for each variable for each instar. The equation of theregression for the mesothoracic legs is y = -0.074 + 0.284 x X (Fs=881.9; df=1 A;r2=.9955). The equation of the line for the metathoracic legs is y = -0.033 + 0.311x X (F5.1004.0; df.=1 A; r2=.9960). The dashed line is the 95% confidence limits ofthe regression line.body mass, and tibial length and diameter within each instar were pooled, anda mean value for each variable for each instar was used in the allometricregressions. These data also indicate that within each instar the metathoraciclegs are approximately twice as long as the mesothoracic legs, while havingapproximately the same diametersFigure 2.5 and 2.6 show the log transformed plots of leg length anddiameter against body mass. Metathoracic tibial lengths followed therelationship of body mass raised to the 0.38 power (SE = 0.01, 1.2 = 0.996),while mesothoracic tibial lengths scaled to mass to the 0.36 power (SE = 0.01,r2 = 0.995) (Fig 2.5.). Metathoracic tibial diameter scaled to body mass to the0.31 power (SE = 0.01, r2 = 0.996), and mesothoracic diameter scaled to the 0.28power (SE = 0.01, r2 = 0.996) (Fig. 2.6.). This indicates that as the animal grows,the limb segments are getting relatively longer and more spindly rather thanmaintaining a constant proportion of length to diameter, as is predicted bygeometric similarity (GSM), or becoming stouter as predicted by distortingallometries (ie. ESM or CSSM). In each case, the slopes of the allometricrelationships between the leg dimensions and body mass were not statisticallydistinguishable between the metathoracic and mesothoracic legs.A convenient index for comparison is the allometric relationship betweentibial length and diameter. The distorting allometries (ie. ESM & CSSM) predictthat lengths will scale to diameters raised to a power less than one, resulting inincreasing stoutness. Geometric similarity predicts an exponent of exactly 1.0or isometry. Locust tibial lengths, however, scale to diameter raised to a power-38-1-0.6 -0.5 -0.4 -0.3 -0.2 -0. 1 0 0.11.40.2Log of Tibiae Diameter (mm).Figure 2.7.The relationship of the log of tibial length and the log of tibial diameterfor mesothoracic (o) and metathoracic (CD legs. Individual points representmeans and standard errors for each variable for each instar. The equation ofthe regression for the mesothoracic legs is y = 0.961 + 1.252 x X (F s=12537.4;df=1 A; r2=.9997). The equation of the line for the metathoracic legs is y = 1.234+ 1.212 x X (F5=35038.2; df.=1,4; r2=.9999). The dashed line is the 95% confidencelimit of the regression line.greater than one ( 1.21, SE = 0.01, r 2 = 0.9999 for metathoracic legs and 1.25, SE= 0.01, r2 = 0.9997 for mesothoracic legs, Fig. 2.7.). These exponents aresignificantly higher (P<0.01 in both cases) than any existing allometric modelpredictions. Thus the tibiae in Schistocerca scale in a manner that is not onlydifferent in value from the existing models, but in direction as well. As was thecase above, the slopes of the allometric relationships for leg morphology werestatistically indistinguishable between the metathoracic and mesothoracic legs.Mechanical Measurements. Figure 2.8•shows a representative spectrumof storage and loss stiffness values across the sampled frequency span. Thereappears to be virtually no frequency dependence to the data. The deviationof the relationship between both storage and loss stiffness and frequency froma slope of zero is significant, but only results in a 6.4% change in actual storagestiffness per decade change in frequency. Results reported by Bennet-Clark(1975) indicated that the jump impulse duration is approximately 25 to 30 ms.Although the impulse is a transient event, and therefore difficult to correlate withsteady state vibration (vis a vis a biologically relevant strain rate), I decided touse the impulse duration as a measure of the half-cycle period and use 22.5 Hz(ie. a 5 point average between 20 and 25 Hz) as my reference frequency forcomparison between samples. The relatively larger scatter of the energy lossdata (E"I) is due to its dependence on what is, in this case, the sine of a smallangle (Eq. 2.3), whereas the energy storage stiffness (E'I) depends on thecosine of a small angle and is smoother (Eq. 2.2). The approximate 30 fold-40--3. 5E'I5 -4.5v.-5.50-6.5-1^0^ 1^2^3Log Frequency (Hz).Figure 2.8.Mechanical test data showing the log of flexural storage stiffness (E'l) andthe loss stiffness (E"I) against the log of frequency of the imposed deformationfor an adult locust of 34 days of age along with their regressions. The equationof the regression for the energy storage stiffness is Y = -3.948 + 0.026 x X(F5=217.7; df=1,67; r2=.7646). The equation of the regression for the energy lossstiffness is Y = -6.238 - 0.085 x X (Fs=19.682; df=1,67; r2=.2271).■■■-3.5117.> -4.5OiJ -5.5• -6.5Ic)c -7.51?-8.50^ 10^20^30^90Age (Days).Figure 2.9.Changes in the log of flexural storage stiffness with age. Data points aremean values with standard error of the mean. Data from individuals on the firstday following a moult are denoted by "s.difference in energy and loss terms results in a resilience of 91% for this data set.The time course of changes in the flexural stiffness (El) of themetathoracic tibiae describes a series of asymptotic curves, where the stiffnessis relatively low immediately after a moult and increases within the next 24 - 48hours by approximately an order of magnitude (Fig. 2.9). This seems to be areasonable consequence of the protein cross-linking and cuticle dehydrationthat occurs during this period (Neville, 1975, Vincent, 1980). The lack of anappreciable decrement in stiffness on the first day of the third instar may reflectan uncertainty in aging the insects (+/- one half day) at a point where thestiffness is changing rapidly, rather than a fundamental difference in thecuticle's behaviour at that point.The fact that the stiffness of the tibiae is relatively low immediately aftermoulting suggested that it might be prudent to create a separate data set forallometric analysis that excluded all stiffness values from individuals on the firstday post-moult, thereby preventing the analysis of the overall scalingprogramme from being biased by the transient physiological events of cuticlestiffening in moulting. Flexural storage stiffness scaled to body mass raised tothe 1.59 power (SE = .02, r2 = .937) for all data, and to the 1.53 power (SE = .02,r2 = .954) for the data without the first-day of instar individuals (Fig. 2.10). Weregard the latter value as characteristic of functionally equivalent states in thepopulation (see discussion) and appropriate for scaling comparisons.A full series of mechanical measurements was not made on themesothoracic tibiae, but morphological measurements indicated that within-43--3—4— 5- 6—7Koue±)— 8.3 —1.5^—1^—0.5^0 0.5Log of Body Mass (grams).Figure 2.10.The relationship of the log of the flexural storage stiffness and the log ofbody mass. Each point is the result of one mechanical test like that shown infigure 8. These data exclude points collected from individuals on the first dayafter each moult. The equation for the regression is y = -4.566 + 1.532 x X(F5=7637.8; df.=1,426; r2=.9442).100 - 75 -50 -25 -L^ •^•0^20 30 30 30 17 20 10 8 10 9 10 9 9 10 10 10 10 10 8 10 9 8 10 10 10 16 15 19 20 20 18 20 17 16 200 :0Age (Days).30^ 40Figure 2.11.The time course of change in resilience of the metathoracic tibia withage. Data are reported as mean values and 95% confidence limit for eachday. Data from individuals on the first day following a moult are denoted by *'s.Numbers listed along the abscissa indicate the sample size for each day.each instar they had virtually the same diameters as the metathoracic tibiae.Thus, one may expect that mesothoracic tibiae will exhibit the same flexuralstiffness as metathoracic tibiae. Mechanical analysis of test pieces of equallengths of metathoracic and mesothoracic tibia from five, 35 day old adultsshowed this to be the case. There was no statistical difference in the meanflexural stiffnesses (t 5 = .9918 < 1 . .0503 = 2.776).The time course of resilience, shown in figure 2.11, demonstrates thatresilience values also increase asymptotically from local minima on the first dayof each instar and achieve higher values over time. Significant differences doexist among means across the entire life history (ANOVA of arcsine transformedresiliences, F5=15.928; df.=34,460). However, it is impossible to say if thisrepresents a real change in mechanical properties, or rather that the smaller,more difficult to handle specimens have a greater variance and therefore lowermean due to the potentially truncated distribution of resilience values (ie.resilience can not be greater than 100%). It is interesting that the mechanicalresilience values for the adults approach an average value of 93%. This is verysimilar to the values of 93% for sheep plantaris tendon (Ker, 1981), 97% for locustresilin (Jensen and Weis-Fogh, 1954) and 91% for the most resilient syntheticrubbers (Ferry, 1980).DISCUSSION The scaling of Schistocerca tibiae results in relatively longer and spindlierskeletal elements, while existing allometric models predict at the least isometry,-46-if not distortions away from increasing spindliness and toward stoutness. Thisempirical scaling relation deviates both from theoretical predictions (McMahon,1980,1984, Bertram and Biewener, 1990) and from some empirical observations(McMahon, 1975, Prange, 1977, Bertram and Biewener, 1990). However, it is invery close agreement with Carrier's (1983) observations for the ontogeneticscaling of limb bones in the jack-rabbit, Lepus. The nature of the existingallometries that predict distortions of morphology is that when beams are longerthan a critical length, with respect to their diameter and their materialproperties, they will either buckle and fail or will deform to a degree that isincompatible with the function of the skeletal system. In the literature it isgenerally assumed that design strategies must increase the mechanical stiffnessof skeletal support structures, presumably by changes in morphology thatincrease the amount of load bearing material, to compensate for increases inthe mass of the structure (eg. ESM or CSSM). Indeed, in one case where limbbones became relatively more slender in ontogeny, producing a potentiallymore deformable structure, there was an observed increase in material stiffnessof an order of magnitude (Carrier, 1983). This makes it important, in the contextof locust tibial scaling, to determine what actually happens to the mechanicalproperties of the leg because the observed morphology predicts the legs arebecoming relatively more deformable as the animals grow.The various scaling models can be used to predict how the flexuralstiffness could scale with increasing body mass. The flexural stiffness has twocomponents; E, the elastic modulus of the material from which the beam is-47-made, and I, the second moment of area--a measure of the distribution ofmaterial across the beam's cross-section. For now we may assume that smalllocusts and large locusts are made of the same material, so that E is a constant(Hepburn and Joffe, 1974b), and changes in El reflect changes in I, which wecan relate to morphology. For tubes of circular cross-section I is proportional todiameter raised to the fourth power. Although locust tibiae and vertebrate longbones do not have strictly circular cross-sections, we will assume cylindricalgeometry, making I proportional to diameter to the fourth power. Given theseassumptions, we can use equation 1, and allometric predictions, to anticipatehow El might relate to body mass. For GSM, diameter (d) scales to mass to the1/3, power and prediction of El follows thus:d oc Mass ."3 ,SOI — (Mass* 333)4, or Mass i333 .Geometric similarity, therefore, predicts that El should scale to body mass raisedto the 1.33 power. Via a similar process, elastic similarity predicts an exponentof 1.50, and constant stress similarity predicts an exponent of 1.60.If one makes this same calculation using the morphological allometricrelationships for the external dimensions reported here for the locust, and makesthe same assumptions about El, one predicts that El in locusts should scale tobody mass to the 1.244 power. This was not the case. The observed exponentfor El as a function of body mass incorporating the complete data set was notsignificantly different from the prediction of constant stress similarity (slope or b-48-= 1.59, is = 0.498, df. = 507). The exponent for the data set excluding first-dayof instar individuals was marginally different from the prediction made by elasticsimilarity (b = 1.53, is = 2.329, df. = 419). Both data sets were different from theprediction of 1.244 power. Therefore, in spite of a morphological programmethat deviates from any expectation, the mechanical properties scale in a waythat is consistent with existing models. This indicates that measurement ofexternal dimensions alone does not provide sufficient information to determinethe mechanical behaviour of the skeleton in this case. This observation posestwo questions: 1) what is compensating for the morphological programme thatallows the mechanics to achieve a mechanically reasonable result, and 2) whatis the design 'strategy' responding to in producing the observed unanticipatedscaling programme?At this point we are unable to resolve the extent to which either E or I orboth are being modulated to produce the observed mechanical scaling. Weare also unable to definitively discriminate the mechanical role of theendocuticle, either within or across instars. However, in chapter three we shouldbe able to resolve the nature of the stiffness or shape changes in the tibia whenwe have examined the scaling of I in the tibiae used to collect the mechanicalmeasurements and determined I for their endocuticular and exocuticularcomponents.Exo- vs. Endoskeletal Design. The sensitivity of I in thin walled exoskeletalsystems to changes in wall morphology suggests a fundamental difference in-49-the use of exo- and endoskeletal designs. Currey (1980, 1984) has generateda model that predicts the optimal inner to outer diameter ratio (k) for a hollowbone. The model plays off the increases in I per unit mass that comes fromusing ever larger diameter, ever thinner, tubular skeletal members against thepenalty of having to carry around the mass of the non-structural material insidethe tube, such as marrow or fat. The analysis predicted different values for kbased on the parameter that the design programme optimized, such as mass-specific strength or stiffness. For a selection of terrestrial vertebrate long bonesthe model seemed to effectively anticipate the relatively thick-walled tubing asan adaptation to resist failure from impact loading (Currey, 1984). A notableexception is the air-filled bones of birds, where the walls are relatively thin, incomparison to the marrow filled bones of terrestrial mammals (Currey, 1984).Figure 2.12 shows the ratio of total mass of bone and luminal contents to themass of a solid bone with the same value of I, for various k ratios. An importantfeature of Currey's model is that the specific gravity of marrow is about half ofthat of bone, ie. the specific gravity ratio (Sg) is 0.50. In Schistocerca the Sg ofthe hemolymph to the cuticle material is about 0.88 (Wainwright et al., 1976).For the locust, the minimum that occurs at about k = 0.35 represents less thana 1% savings in mass over a solid rod morphology (Fig. 2.12). Measurementsfrom the cross-section of the metathoracic tibial segment in a 10 day old adultSchistocerca from Jensen and Weis-Fogh (1962) show a k 0.92, indicating thatthis skeletal member is not approaching the optimum associated with aminimum mass.Figure 2.12.Theoretical model of optimal internal to external diameter ratio (k) forhollow skeletal structures with various ratios of lumenal content to wall materialspecific gravity (Sg). This formulation describes beams that all have the sameI. The line Sg = 0.88 is the condition that exists in Schistocerca gregaria whichhas a slight minimum at k = 0.35. Optimizing for mass specific stiffness, or impactresistance in the locust produces even shallower minima at lower k values. Fora Sg equal to 0.0, which approximates the air-filled bones of birds, the modelpredicts thin walled tubes limited by buckling. As one moves to relatively lessdense tubing material, the minima become shallower and predict thicker walls.Sg equal to 0.5 is the condition describing bone (Currey, 1980, 1984), which hasa shallow optimum at k = 0.7. At Sg = 0.13 (steel tubing filled with an aqueousmedium) there is a deep minimum at a k of 0.93 with a weight savings of about51%. The boundary condition of equal specific gravities for the tubing and theinside contents (Sg = 1.00) predicts that the minimal mass solution occurs in solidrods rather than hollow tubes. The dashed vertical line originating at k=0.98 isthe limit to k imposed by unstable buckling based on data for the materialproperties of locust cuticle (Vincent, 1980) and the analysis of buckling in Currey(1980). This represents the ultimate limit to being thin walled.The fact that insects do not have skeletons made from thick-walled tubingmay indicate that minimizing the mass of non-structural material is not the over-riding design strategy. In exoskeletal animals, where the material on the insideof the tubing is the muscle and circulatory fluid, the added weight of thismaterial may not represent the same penalty as marrow or fat does invertebrates. If it is the locust's strategy to maximize its internal volume, thenfigure 2.12 predicts a thin-walled morphology that achieves large volume thatis limited only by buckling (Currey, 1980). The dashed vertical line in figure 2.12originating at k = 0.98 indicates the limit imposed by unstable buckling in abeam constructed from locust cuticle. The similarity of the observed k ratio andthe buckling limit would seem to support the idea that the design in locustsmaximizes internal volume.Scaling in Cursorial & Jumping Insects. The observation that locusts'limb morphology deviates from the existing allometries while other exoskeletalanimals like cockroaches and spiders do not (Prange, 1977), would seem toindicate that the exoskeletal body plan does not in itself determine the scalingof limb dimensions. That is, the exoskeletal design may have allowed thisoption, but evolution has not demanded that all such designs follow it.Therefore, it would seem reasonable to suggest that the developmentalprogramme that produces the morphological scaling in Schistocerca representsan adaptation for some specific functional attribute. I suggest that thekinematic and energetic demands of jumping in growing animals may be met-53-more effectively by the allometry seen in the locust, rather than thegeometrically similar growth pattern seen in cursorial animals like the cockroach.If it is true that the morphology of the metathoracic legs is demonstratinga specialization for jumping, then we might hypothesize that the pro- andmesothoracic legs (ie. 'walking legs') would scale in a manner different to themetathoracic leg ('jumping leg"), but this is not the case. It has been notedthat prothoracic legs appear to follow isometry (Gabriel, 1985a); however, thedata on mesothoracic legs clearly shows a similar developmental programmeto the metathoracic legs. While maintaining the same scaling relationship tobody mass across ontogeny, the metathoracic tibiae are twice the length ofthe mesothoracic tibiae. The third power dependence of deflection of aloaded beam on length (equation 2.1.) predicts, therefore, that themetathoracic tibiae are eight times more deflectable than the mesothoracictibiae given the same flexural stiffness. Since equal lengths of mesothoracicand metathoracic tibia have the same flexural stiffness in adults, it seemsreasonable to assume that the interleg similarity in stiffness per unit length wouldbe maintained across ontogeny. Further, the peak loads during jumping areapproximately 20 times the force of gravity (Bennet-Clark, 1975) and aredistributed over only two jumping legs, whereas the loads of standing or walking(ca. 1 x the acceleration of gravity (g)) are distributed over six legs. The walkingtibiae, then, may experience several hundred times less bending deflection inwalking than are the metathoracic legs during jumping. Even in the casewhere a locust lands on a single mesothoracic leg at the end of a jump, thekinetic energy would be absorbed by a single leg with one eighth of thecompliance of each metathoracic leg. In this extreme case the mesothoracicleg would still experience four times less deflection than a metathoracic tibia.Even though the mesothoracic tibiae are probably not experiencing exclusivelya bending load, this analysis indicates that they are distinctly over-built. Thus,the genetic constraint on the development of metameristic, morphologicalcharacters may limit the ability of natural selection to minimise the amount ofextra material the animal carries around. It may turn out that the geneticmechanisms that control metathoracic leg development are linked functionallyto the genetic control of mesothoracic legs and even perhaps to appendageson other segments of the body. Interactions across body segments, withrespect to the control of expression of metameristic characters, have been seenin the bithorax gene complex in Drosophila (Lewis, 1978). Since jumpperformance, in terms of take-off velocity, is inversely dependent on body mass(Bennet-Clark, 1977), there would seem to be a benefit in keeping excessweight to a minimum. If so, then the overdesign of the mesothoracic legs mayrepresent a penalty to be paid for the sake of having jumping legs that performwell and having a genetic control mechanism that is not specific to a singlebody segment. It may prove interesting to test this hypothesis by examining thescaling of antennae or mouth parts of the locust to determine the extent of thisinteraction.Ontoaenetic vs. Phvloaenetic Scaling. In this study I have used ontogeny-55-as a model for the effect of body mass on the morphological and mechanicaldesign of the limb skeleton of Schistocerca. I found a developmentalprogramme that results in increasingly spindly legs, suggesting more easilydeformable beams, but the mechanical properties are adjusted to produceelastically similar tibiae. I believe that this provides information about themechanical design of jumping animals, but is ontogeny a justifiable model forthe effect of body size? Is it fair to assert that the ontogenetic scaling observedin locusts demonstrates mechanical design principles? If it is, then we wouldexpect to find similar scaling in other jumping animals.It seems significant that the morphological scaling relationship reportedhere, while different from phylogenetic comparisons, is so similar to theontogenetic morphological changes seen in the jack-rabbit (ie. tibia length —tibia diameter 121 for locust, tibia length a tibia diameter13° for Lepus, Carrier,1983). Also interesting is Carrier's observation that material stiffness and secondmoment of area of the metatarsal bones increase with the first power of bodymass. This indicates that the flexural stiffness is increasing with the second powerof body mass, faster even than the 1.53 power observed for locusts.It is tempting to suggest that the similarity in Schistocerca and Lepusontogenetic scaling is demonstrating a developmental strategy that is adoptedgenerally by jumping animals. However, there is an important difference in thedevelopment of locusts and jack-rabbits that points to a potential danger inusing ontogenetic observations to examine the effect of body size. Thisdistinction lies in comparing animals that are not functionally similar. Both adultlocusts and adult jack-rabbits hop or leap as their mode of locomotion, butwhereas juvenile locusts hop (Gabriel, 1985a), juvenile jack-rabbits prefer not to(Carrier, 1983). Carrier (1983) has reported that the neonate jack-rabbits (<300grams) are 'unsteady' locomotory performers, relying on crypsis to avoidpredation. Over 600 grams, however they have developed good locomotorperformance, and readily resort to jumping when startled. Whatever thepressure is that has resulted in the developmental programme in jack-rabbits,it is not the need to be a functionally adequate jumper over the entireontogenetic range of body size. A jack-rabbit achieves high locomotorperformance over only the final four fold range in body mass, while the locustachieves high performance, in terms of distance covered, at points within eachinstar over a 200 fold range in mass (Gabriel, 1985a). Interestingly, locusts starteach instar as poor jumpers (Gabriel, 1985a, Queathum, 1991) with relatively softcuticle (Fig. 2.9). Over the following 24 to 48 hours the cuticle stiffens byapproximately an order of magnitude and their locomotor performanceimproves. Thus, it appears that locusts go through developmental changeswithin each instar that are similar to those that the rabbit goes through over itsentire lifetime. In a sense, the rabbit's development is composed of a single"instar", while the locust's development is composed of functionally competentindividuals that come in six different sizes. Since the post-24 hour individuals ofeach instar represent functionally similar locusts of different sizes, I feel confidentthat the scaling relationships reported here are providing important insights intothe design of jumping animals.-57-Design in Jumping Tibiae. It is appropriate, therefore, to attempt toexplain the observed skeletal scaling in the context of the locust's locomotorstrategy. As Bennet-Clark (1977) points out, as animals get bigger theiracceleration in jumping decreases, so that dynamic, mass-specific loading injumping is decreasing at the same time that the relatively static accelerationsencountered in standing or walking are increasing. The increasing slendernessof the tibiae may be a response to falling accelerations produced in the jumpof large animals compared to small ones. If larger locusts produce lessacceleration than small ones, then their longer limbs may be an adaptation toincrease the time that their feet are able to do work on the ground. Adultlocusts of 3.5 grams body mass produce approximately 20 g's of acceleration(Bennet-Clark, 1975), while adult fleas (Spylopsyllus cuniculus) of 0.45 milligramsproduce accelerations of over 135 g's (Bennet-Clark & Lucey, 1967). Locuststibial morphology may be following a programme that is designed toaccommodate declining accelerations encountered with increasing body mass,rather than body mass per se. Therefore, scaling relationships that areattempting to explain the mechanical design programme relating morphologyand body mass make predictions in a direction opposite to that seen in locusts.Counter to this argument, however, Scott and Hepburn (1976) have suggestedthat small locusts do not produce larger accelerations than adults. They reporta constant relationship of approximately 10 g's of acceleration in severalAfrican grasshoppers, as well as through the ontogenetic sequence of Locustamigratoria. None of their observations seems to be as large as the 20 g's-58-reported by Bennet-Clark (1977) for adult Schistocerca gregaria. It would,therefore, be of value to know explicitly how the accelerations developedduring the jump change from small to large locusts.An alternate explanation for the morphological programme may lie in are-interpretation of the function of the exoskeleton. Schmidt-Nielsen (1984)suggests that skeletons are rigid structures that act as either support beams orlever arms to be acted on by muscles to provide movement. Two observationsin this study indicate that perhaps the locust tibial skeleton is acting not so muchlike a rigid lever arm, but rather as an elastic energy storage device. Fromequation 2.1 (arreanged to describe a cantilever) and a peak acceleration of20 g's (see above), the ground reaction force for a 0.003 Kg adult locust willproduce a deflection at the end of the tibia on the order of 3 mm, or about13% of the tibia length. In fact Brown (1963) has stated that some of his highspeed films show the tibiae bending during jumps. The energy required todeform a linear spring is equal to one half of the product of the deformationand the force applied, which in this case is 1.2 mJ of energy for both legstogether. If we incorporate a cuticular resilience of 92%, then approximately 1.1mJ of energy are returned as elastic recoil from the tibiae during the jump.Bennet-Clark (1975) has reported that an adult female Schistocerca gregariarequires about 11 mJ for a jump. Thus approximately 10% of the total energyof the jump is recovered from energy stored in the tibiae. The relative increasein tibial spindliness with increasing size may represent an attempt to create amore deflectable beam, and therefore, a larger capacity energy reservoir, as-59-the peak accelerations are falling during ontogeny. Additionally, the highresilience values of 90 - 93% for tibial cuticle (Fig. 2.10) indicate that thematerial's properties are well matched to an energy storage function.These observations suggest that it is more appropriate to think of locusttibiae as bending springs rather than simple rigid levers. This use of energystorage seems significantly different from other systems previously examined inthat it does not act as a momentum collector, capturing kinetic energy from amuscular contraction in a previous stride as potential spring energy to berecovered as kinetic energy in the following stride. Instead the spring energyis stored and recovered in the same stride, not unlike the model proposed forprimary flight feathers in pigeons (Pennycuick and Lock, 1976). Why then storeit when mechanical hysteresis will only decrease the amount of muscular energythat does any useful work in locomotion? It may turn out that the energy storedin the tibia early in the force impulse in the jump is stored at a time when themechanical advantage of the muscle-apodeme-tibia lever system is high, butthe ability for the locomotor system to do work on the ground is low, andenergy can be stored in the spring. That energy could then be returned laterin time within the same loading event when the mechanical advantage of themuscle-apodeme-tibia lever system is low (Bennet-Clark, 1975), but the abilityto do work is high. A useful metaphor for this type of design may be an archer'scompound bow where eccentric cams alter the mechanical advantage of thebow on the arrow. When a loaded compound bow is released the forcecontinues to rise as the arrow accelerates, producing higher velocities andlonger distances than a traditional bow where the force falls as the arrow isaccelerated. It may prove that the tibial-springs are utilizing changes inmechanical advantage in the same manner, maximizing take-off velocity andas a result trajectory distance. This possible basis for a design strategy will beexamined in more detail in chapter five.CHAPTER 3.SCALING MODULUS AS A DEGREE OF FREEDOM.INTRODUCTION In chapter two we learned that the scaling of external dimensions of thelocust's legs does not predict the scaling of the mechanical behaviour of thelegs in bending. The flexural stiffness of the tibiae scale in a manner that issimilar to predictions based on the elastic similarity model, but the legs' externaldimensions do not. So there is an 'uncoupling' between the morphology andthe mechanical properties of the legs. It was speculated that this uncouplingcould be the result of two events. Either the second moment of area (I) of thelegs could be related to the external dimensions in ways that are not obvious,or the material stiffness (E) of the cuticle could be changing to provide anadditional degree of freedom in the design of the skeleton in accommodatingincreasing body size.With respect to compensation, it is possible that the modulus of thecuticle material is altered to accommodate changes in morphology of the limbsegment in order to maintain an elastically similar flexural stiffness. Certainly, themodulus increases during the immediate post-moult period of scleritization dueto dehydration (Hepburn and Joffe, 1974a, Vincent and Hillerton, 1979, Vincent,1980). Indeed, it may be that the material properties of the cuticle are differentat different ages. Hepburn and Joffe (1974b) have suggested, however, thatthe cuticle of Locusta migratoria maintains a similar stiffness in tanned fifth instarand adult femoral cuticle. They suggest that the ratio of stiffness to mass is aconstant for tanned cuticle across instars, and that this is a response to a-62-constant ratio of load developed in a jump to body mass across the instars(Hepburn and Joffe, 1974b). If the tanned cuticle of Schistocerca has the samemodulus independent of age, then the design strategy that results inincreasingly spindly legs in locusts must be fundamentally different than thatobserved for Lepus.It is also possible that the distribution of cuticle material changes acrossinstars, resulting in a change in I that is not reflected in a change in theexternally measured diameter. The formula for I is('max in tensionI y2 ciA (3.1)J, Y max in compressionwhere dA is the increment of cross-sectional area located a distance y awayfrom the neutral axis, a line through the centre of mass of the cross sectionnormal to the bending moment (Gordon, 1978) (Fig. 3.1). The importance of thisrelation is that material away from the neutral axis contributes greatly to thestiffness of the test piece. Because the cuticular segments of locusts are thinwalled cylinders, a small adjustment in the distribution of material on the insideof the cross-section of a locust leg could impart large changes in mechanicalproperties that are not easily inferred from measurement of external dimensions.The sensitivity to changes in internal material distribution is not necessarily truefor the thick-walled bones of terrestrial vertebrates. Certainly, endocuticle isadded after moulting within each instar (Neville, 1975), although it is not clearif material is added in a manner that would alter the relationship between theleg diameter and I. It is also not clear that accumulations of endocuticlecontribute to changes in the mechanical properties of the tibiae. At apolysis-63-Figure 3.1.A diagram of a beam of irregular cross-section to show the calculationof the second moment of area (I). The dotted line shows the neutral axis forbending loads applied in the vertical direction. The neutral axis passes throughthe centre of gravity, or centroid, of the cross-section. I is calculated as theintegral of the incremental amounts of cross-sectional area (dA) weighted bytheir distance from the neutral axis squared (y2). This calculation was performedon video images by first identifying the centroid of the section, then summingthe number of super-threshold pixel elements in each raster line weighted bythe distance of that line from the centroid in raster lines squared. These valuesfor each raster line were then summed across all of the raster lines in the cross-section.• Neutral Axisy max in tensiony 2dAy max in compression(ca. 1-2 days before moulting, Queathum, 1991) there is a reorganization of theendocuticular material that involves an enzymatic digestion of the endocuticle(Zacharuk, 1976), presumably decreasing the second moment of area.However, I observed no decrement in flexural stiffness within any instar that canbe correlated with the occurrence of apolysis (Fig. 2.9).The accumulation of endocuticle also does not explain how changesmight be mediated across instars. Gabriel (1985b) reported that themetathoracic tibia's cuticle thickness in the anterior direction relative to thelateral direction maintains a constant proportion across the juvenile instars, butincreases by 50% in adults. This differential thickening could contribute tochanges in I that are not reflected in external diameter. However, it is notknown to what extent endocuticle accumulation is responsible for the increasewall thickness seen in adults, or why the discontinuity in wall thickness betweenall of the juvenile instars and the adults is not reflected in the flexural stiffness.What Gabriel's data do suggest is that if I is being adjusted to maintain theobserved relationship between El and body mass, it is not, in juvenile instars atleast, being accomplished with simple changes in wall thickness. Rather, thereare probably changes in cross sectional shape that are producing changes inI.In this chapter I examine this uncoupling explicitly by measuring thesecond moments of areas of the same legs that I tested in chapter two andtease apart the relative contributions of material stiffness (E) and distribution ofmaterial (I) to the scaling of flexural stiffness (El).METHODS AND MATERIALS.Specimens used in this study were a sub-sample of the same specimensused in chapter two. Therefore, methods for animal husbandry, morphologicalmeasurement and mechanical testing are exactly the same. Specimens formeasurement were chosen to represent the range of body masses covered inthe life of the locust. Twenty-one first and second instars were chosen torepresent small locusts, seven fourth instars were used to provide intermediatesamples, and sixteen adults were used to provide values for the large animals.Additionally, fifty-six fifth instar samples were used to show how the changes insecond moment of area occur within an instar. Individuals from the first day ofeach instar were eliminated from the analysis of scaling to make comparisonsonly between individuals that were felt to be functionally similar.Following three-point mechanical testing described in chapter two, testpieces were placed in 37% formaldehyde solution for fixation and stored forlater embedding and sectioning. Each tibia was dehydrated in an alcoholseries and embedded. The technique used to produce cross-sections of thelocust legs was different in the large and the small specimens. First and secondinstars' legs were embedded in paraffin and sectioned with a microtome at athickness of 20 gm. Older locusts' legs were embedded in araldite. Eacharaldite block was sectioned at 800 pm with a bone saw. These 800 pm thicksections were epoxied to a glass slide and thinned to 0.005" (1.27x1 0-4 m) withalumina sand paper of increasing grit size. The surfaces were then polished with600 grit polishing paper. Optical imperfections in the surfaces of the specimens-67-were filtered out by placing a drop of immersion oil on the section and thencovering with a glass coverslip. Sections used for measurement were chosenfrom the approximate mid-shaft location on the tibia and had no distortions ofprofile associated with spurs. The cross-sections of the tibia were observed witha compound microscope fitted with a video camera. The labour intensivenature of this process resulted in the relatively small subset of the legs tested inchapter two being used in the analysis of second moment of area.Images of the legs' cross-sections were captured from the video signalwith a video frame grabbing computer interface (PIP 1024B, Matrox ElectronicSystems Ltd., Dorval, Quebec, Canada) and stored in computer memory.Editing and analysis of the video images was accomplished with the V videoimage processing system (Digital Optics Ltd., Aukland, New Zealand) on a PC-type computer. Each 256 greyscale level video image was transformed into abinary image by manual detection of the perimeter of the entire cuticle andthen the exocuticle alone and then setting a threshold greyscale level thatdefined the leg cross-section against the background. The perimeter of theendocuticle was produced by arithmetically subtracting the image of theexocuticle from that of the entire cuticle. The V software then integrated eachof the video images to calculate the second moment of area of the crosssections that represented the entire cuticle, the exocuticular component, andthe endocuticular component for each leg that was successfully sectioned.Measurements of tibiae diameters of tibiae sections showed less than a1% difference from values reported for the same specimens in chapter two.-68-Dimensional changes that resulted from the histological process were, therefore,ignored.The image analysis software computes the second moment of area withthe following summation:= Ey Ex (y-y°)2 dAThe image analysis software performs this integration numerically as the sum ofpixel elements of a defined grey-scale level, weighted by the square of theirdistance from the neutral axis in raster lines for each raster line (E r). The valuesare then summed over all the raster lines from the location of the centroid outto the margin of the section (E y). Also mentioned previously, material at themargins of the cross section contribute greatly to the stiffness of the test piece.As such, orientation of the non-circular cross-sections, both in the mechanicaltesting and in the measurement of I, can have significant effects on the results.To control for the role of orientation, a sub-sample of images were printed asbinary images. This sub-sample of images was recollected with the video imageanalysis equipment and I was calculated as the neutral axis was rotatedthrough a series of orientations from +90° to -90°.RESULTSWithin Instar Changes in I^The time course of changes in flexuralstiffness for the sub-population of locust legs used in this study are demonstratedin figure 3.2. E'l averaged a low value of 9.78x10 7 Nm2 on the first day of theinstar, and increased by about 30 fold by the second day. Thereafter, E'l-69- -3.5-4.5-6.5-7.5 F-1 2^3^4^5^6^7Age (days within fifth instar).Figure 3.2.The relationship of the log of flexural stiffness and age within the fifthinstar. Data points are mean values and standard errors of the mean. Age isreported as days within the instar. -13-14-16-17 r1 3.^2^3^4^5^6^7Age (days within fifth instar).Figure 3.3.The relationship of second moment of area (I) for the exocuticular (x),endocuticular (o) and entire cuticle components (0) and age within the fifthinstar.remained relatively constant at a value of approximately 4.13x10 -5 Nm2 .The relative contributions of the cuticular components are shown in figure3.3. Values of I for the exocuticle alone had a relatively high mean value of3.60x10-15 m4 on the first day of the fifth instar and remained close to this valuefor the remainder of the time spent in this instar. The endocuticularcomponent started out at a low mean value of 1.95x10 -16 m4, butincreased asymptotically until approximately day five where it attained a meanvalue of 8.61x10 -15 m4. The values of I for endocuticle on the first day of the fifthinstar had a large variance as one of the samples had almost no measurableendocuticle. Examination of the actual cross-sections of the tibiae suggestedthat immediately after the moult there was no measurable endocuticle, andthose samples that were taken on the first day that contributed to the non-zeroestimate of endocuticular I reflected the uncertainty in measurement of age(+/- one half day).Significantly, there appeared to be no detectable breakdown of theendocuticular material that could be correlated with apolysis on or about dayfive. However, it was observed that after day five there was a clear separationof the endocuticle from the underlying epithelial layer. Indeed, in some samplesit was also possible to observe the next stage's tibia folded up within the lumenof the tibiae, indicating that the events that are associated with apolysis andthe production of the next stage's exoskeleton had occurred without reducingthe second moment of area of the endocuticular material.The time course of changes in I for the entire cuticle is also plotted infigure 3.3. As this parameter is in fact the arithmetic sum of the I calculated foreach of the components, it demonstrates a small increase from a mean valueof 4.38x10 -15 m4 on day one to a mean of 1.22x10 -14 m4 on day three and then-72-remains fairly constant for the remainder of the instar.Scaling In the sample of locust legs used in this chapter the flexural stiffnessof the legs scaled to body mass raised to the 1.505 power (SE = 0.030, r 2=0.966).This slope is not significantly different from the slope of 1.532 (t5=0.342, df. = 513,p<0.05) for the entire sample reported in chapter two. This indicates that thedata used in this analysis represent the population as well as do those inchapter two.Figure 3.4 shows the scaling relationships for the second moment of areafor the various components of the cuticle with increases in body mass. I of theexocuticular component of the tibiae scaled to body mass raised to the 1.090power (SE = 0.026, r2=0.952). I of the endocuticular component scaled to bodymass raised to the 1.258 power (SE = 0.031, r 2=0.949). I, calculated for the entirecuticle, scaled to body mass raised to the 1.195 power (SE = 0.027, r 2=0.957).By normalizing the measured value of El from chapter two by thecalculated value if I for both of the cuticular components we can estimatewhat the modulus of elasticity, E', is for the tibiae. Figure 3.5 shows the scalingrelationship of E' with increases in body mass. E' scales to body mass raised tothe 0.311 power (SE = 0.033, r2=0.511). This slope was significantly greater thana slope of zero (t5=9.541, df.=89, p>0.05).Values of E' ranged from a low value in first instars of 3.9x10 8 N/m2 to ahigh value in adults of 1.6x10 m N/m 2. The estimated value of E' for adults is verysimilar to the value of 9.4x10 9 N/m2 (960 kg/mm 2) reported by Jensen and Weis-Fogh (1962).Orientation of Neutral Axis^The shape of the tibia cross-sections are not-73--2 -1.5 -1 -0.5 0 0.5 1-13-14-16-17Log of Body Mass (g).Figure 3.4.The relationship of the log of second moment of area (I) for theexocuticular (x), endocuticular (o) and entire cuticle components (El) and thelog of body mass. The equation of the regression of exocuticular I on bodymass was y = -14.569 + 1.090 x X (F5 = 1738.74, d. = 1, 89, r2 = 0.9523). Theequation of the regression of endocuticular I on body mass was y = -14.283 +1.258 x X (F5 = 1611.19, df. = 1, 89, r2 = 0.9488). The equation of the regressionfor the entire cuticle's I on body mass was y = -14.092 + 1.195 x X (F 5 = 2016.63,df. = 1, 89, r2 = 0.9568)..o 08.5 L--2^-1.5^-1^-0.5^0^0.5^1Log of Body Mass (g).Figure 3.5.The relationship between the log of tensile modulus (E') and the log ofbody mass. The equation of the regression E' on body mass was y = 9.596 +0.311x X (F, = 91.040, d. = 1, 89, r2 = 0.5113). E' was calculated for the entirecuticle treated as a homogeneous and continuous structure, see text fordiscussion.10.5Figure 3.6a.Example cross-section from a fifth instar locust meta-tibia. The top of thepicture is the anterior direction. For scale, the diameter of the section in theanterior-posterior direction is 2.523 mm.Figure 3.6b.Example of section shown in figure 3.6a processed for calculation of I forthe entire cuticle. The inner and outer margins of the image in figure 3.6a havebeen defined manually, and the area of the section has been assigned a singlegrey-scale value. The image analysis software then performs the calculation onthe image, summing over all the pixels that are of a pre-defined grey-scalelevel.Figure 3.6c.The same section as in figure 3.6a & b with the margins defined only forthe exocuticle.Figure 3.6d.The same section as in figure 3.6a, b & c with the margins defined only forthe endocuticle.43C:, Ucircular or even elliptical. They form a very characteristic shape that impartspotentially important mechanical behaviour. Figure 3.6a shows a representativecross-section from a fifth instar locust's tibia. The relatively circular, posteriormargin and relatively thicker walled corners on the anterior laterai margins arevery characteristic. The necking that produces the concavity at approximatelythe one quarter to one third point from the anterior to the posterior margin co-occurs with a membranous connection that spans the lumen of the tibia.Immediately after a moult when the cuticle is relatively soft, this membrane mayprovide a tensile stay that prevents the tibiae's cross-sections from becomingcircular until the cuticle hardens.Figure 3.7 shows how the value of I for the cross-section of six legschanges as the neutral axis is rotated through a range of angles from +90° to -90° . Individual legs show a local minimum of I along the anterior-posterior axis,with I increasing by as much as 3% with a rotation of 5° to either side. Theaverage behaviour shows a more modest increase in stiffness as the neutral axisis rotated through + 5° of 0.4%. The average values for I only drop by anaverage value of 4.54% when rotated through + 20° . This is contrasted with anellipse of similar aspect ratio where I falls by 6.40% (Fig. 3.8).So far, I have talked about cross sections that were free of anymorphological feature that was associated with spurs, but spurs do exist andtheir contribution to the mechanical behaviour of the tibiae must beconsidered. The presence of spurs produces a bilaterally asymmetrical distortionin the relatively circular perimeter of cuticle along the posterior margin of thetibiae cross-sections. The distortion takes the form of a bump that puts morematerial away from the neutral axis on alternating sides of the tibiae.Presumably, this distribution of material increases I and shifts the orientation of-78-0.4-100 -80 -60 -40 ..c.,^0^4060 io^100Angle of Orientation of Neutral AxisFigure 3.7.Plot of the relationship between I and the orientation of the neutral axisfor a sample of six legs (■) and their average value at each orientation.0.4-100 -80 -60 -40^ io ^80 100Angle of Orientation of Neutral AxisFigure 3.8.Comparison of the effect of orientation of neutral axis on I for an exampleleg and an elliptical cross-section of similar aspect ratio.Figure 3.9a.Example cross-section from the same leg as shown in figure 3.6a with thebase of a spur projecting from the posterior margin of the section.Figure 3.9b.Example of section shown in figure 3.9a processed for calculation of I forthe entire cuticle in the same manner as in figure 3.6b.Figure 3.9c.The same image as in figure 3.9b, but flipped around its centre of gravityso that a mean influence of the spurs on I on both sides of the leg could beestimated.11 :1 . I: . 14^411tCO^,1 ion.'^ .^••Lp.ton?ft.'-34, •• VOIOSI^.•I .•• I^• •^.46^4 4h^"•;^''Y..•11) '^.••••••./•••■••01.11 -IP<-5 0.9-cg 0.8-21 0.7-co.>. 0.6-c5a)cc 0.5-0.4-100 -80 -60 -40 -20^O^io^40^60^80 100Angle of Orientation of Neutral Axis.Figure 3.10.Plot of the relationship between I and the orientation of the neutral axisfor an average leg with spurs on both sides. Data calculated for eachindividual cross-section (dotted lines) are superimposed on the averagebehaviour (solid line) of a leg with spurs on alternating sides along its length.-83-maximum stiffness toward alternate sides of the tibiae. This should result in anaverage behaviour that has a deeper and broader local minimum of stiffnessalong the anterior-posterior axis. Figure 3.9a is a figure of a tibial section witha spur on one side. Figure 3.10 is the relative I of a compOsite sectiongenerated by reflecting the section in figure 3.9b through its centre of gravityalong the anterior-posterior axis. The neutral axis is then rotated through thesame range of orientations as figure 3.7. The composite behaviour shouldrepresent the average behaviour of the tibia along its length. This plot showsthat there is a modest local minimum of stiffness at the anterior-posterior axis.This might suggest that the spurs play an important mechanical role inamplifying any built-in stability to the loading of the tibiae in bending.DISCUSSIONWithin Instar Changes in I The data on how the values of I changewithin the fifth instar seem somewhat equivocal on the question of the relativecontributions of the two cuticular components toward whole tibia stiffness. Overthe first three days the rapid increase in endocuticle matches the rapid increasein flexural stiffness. If the material stiffness is constant over that period theseresults would suggest that the stiffness demonstrated by the whole tibia resideslargely in the endocuticular component. Over the final five days of the instarthe flexural stiffness is more reflective of changes in I in the exocuticle,suggesting that the exocuticle is making the most significant contribution toflexural stiffness. The data of Hepburn & Joffe (1974a) indicate that during thefirst 24 to 36 hours the cuticle, which over that time period is almost entirelyexocuticle, increases in modulus by almost an order of magnitude. If so thenwe can not really depend on the correlation between I and El to provide-84-information on the relative contributions of the cuticular components to tibialstiffness. From information provided by Zacharuk (1976) and Queathum (1991)on the effect of apolysis on the endocuticle, it was anticipated that changesin the endocuticle would be uncorrelated with changes in El (see introductionthis chapter). The fact that this was not the case prevents me from being ableto discriminate the contributions of the cuticular components based on my dataon fifth instar locusts. For this reason, in estimating the elastic modulus I willconsider the exocuticle and endocuticle as homogeneous and continuouscomponents as did Jensen and Weis-Fogh (1963). As a result the modulusvalues that are estimated represent lumped parameters for entire cuticle, andthe stiffness for each component may be different.Scaling In chapter two we observed that if I is a function of the fourthpower of the diameter of the limb segment, then I should scale to body massraised to the 1.244 power. Our observation that I for the entire cuticle scales tobody mass raised to the 1.195 power is not significantly different from thatprediction (ts= 1.862, df.= 91 ,p < 0.05). This provides an answer to our originalquestion of the nature of the observed uncoupling of morphology andmechanical properties. The locusts are achieving elastically similar flexuralstiffness by scaling the material stiffness of the cuticle, rather than applying anallometric scaling to I by changing the relative distribution of material in thecross section. This conclusion is reflected in figure 3.5 where E scales to massraised to the 0.311 power.Of all the scaling relationships reported in this thesis, the relationshipbetween E' and body mass has the poorest statistical strength. This is perhapsnot surprising, as the variance in these data are the product of variability in the-85-mechanical measurement of El and the variability in estimating I. Still therelationship is highly significant (Fs = 91.04, df. = 1,89) as a descriptor of the data.Therefore, I feel reasonably sure that the elastic stiffness of the cuticle materialis being treated as a scaled commodity in the design of the locusts' legs.Because the modulus is changing in functionally similar hoppers thescaling of modulus for these legs seems different from that for Jackrabbit bones(Carrier, 1983). In the locust the modulus is scaling to provide an additionaldegree of freedom in adjusting morphology to accommodate increasing loadswith increasing size. Given a design strategy that tries to keep a constantdeformation per unit length with increasing size, the non-mineralized skeleton ofthe locust can adopt morphology that is not predicted by the elastic similaritymodel. So now we know how the uncoupling between morphology andmechanical behaviour is accomplished, but it remains to be seen why it isuncoupled. Indeed, this may presuppose that the morphology demonstratedin the locust leg is an adaptation to a specific design issue. In chapter five I willoffer three hypotheses for why the locust might have adopted the scaling of themorphology of its legs that it did.I do not know the nature of the changes that I have observed in E'. Thecuticle is a fibre-reinforced composite material (Neville, 1975) and there are anumber of ways these materials can be modified (Wainwright et al., 1976). Itwould be interesting to investigate whether the animals are altering the proteinpolymer that glues the chitin crystalline fibres together, by altering hydration orcross-link density perhaps, or if the crystalline fibres are themselves altered, bygeometry (ie. aspect ratio) or volume fraction for instance.For the endocuticular and exocuticular components of the cuticle I isscaled to body mass differently. I have treated the cuticle as homogeneous in-86-material properties resulting in a preliminary conclusion that the modulus is ascaled commodity. An alternative hypothesis is that the modulus for eachcomponent is different and scale independent, and that the amount of eachcomponent is varied to produce the average behaviour that is observed forentire tibiae. If this were true, we could estimate how different in modulus thecomponents would have to be to produce the lumped behaviour. I believe thisalternative hypothesis can be rejected by considering the limits to this designstrategy. The hypothetical limiting cases are where one or the othercomponent has a stiffness of zero and the scaling of the average system isstrictly a reflection of the scaling of I of the non-zero stiffness component.Specific to this case we need to know if we assume moduli for the cuticularcomponents, the inner of which's I scales to body mass 1258 and the outer ofwhich's I scales to mass' °90, can we get the entire structures El to scale tomass 1.53? The answer would seem to be no. Even if the modulus of theexocuticle were zero, the entire cuticle's El would only scale to body mass 1258 .Therefore, even if we accept the assumption of zero stiffness for the exocuticle,which seems unreasonable, we are forced to conclude that the modulus of thecuticular components is altered in response to changes in body size.Orientation of Neutral Axis The observation in figure 3.7 & 3.8 that orientationhas so little effect on the value of I near orientations around the anterior-posterior axis suggests the interesting possibility that the cross-sectional shapeof the tibiae provides a built-in stability to the limbs. One can imagine thatsome fraction of the jumps that a locust makes in its lifetime are made offmorphologically complex surfaces. As a result it is also possible that the loadsencountered during the jump impulse will not be applied collinearly with the-87-anterior-posterior axis. If the tibiae has a more elliptical cross-section, then anyoff-axis loading will tend to flex the leg in the medial lateral axis—a direction inwhich I, and therefore relative stiffness, is reduced by approximately 50%.However, the marginally greater roll off with angle of rotation of I for the ellipserelative to the locust leg demonstrates the subtle degree of this stability.The influence of spurs on the flexural stiffness of the tibiae may be moresignificant than the composite behaviour estimated in figure 3.10 might suggest.Individual sections with spurs have no local minimum of stiffness around theirmaximal stiffness values which themselves are oriented about 25° off of theanterior-posterior axis. At each segment of the length of the leg that has a spurthe primary loading will be off the axis of greatest stiffness. As a result each littlepiece of leg will be subject to flexion in the direction of the anterior-posterioraxis, lateral to its direction of greatest stiffness. The magnitude of the flexion inthis small length of leg will itself be small because of the large dependence ofdeflection in a beam on length (Eq 1.8). Further, distal to each small, laterallyloaded segment of leg will be a similar small segment loaded in the oppositedirection because it has a spur on the other side. It may turn out that thisalternating pattern of laterally flexed, short beams produces greater stabilitythan the composite section will suggest.I would prefer not to draw too strong a conclusion from the analysis of theshape of this length-averaged or composite cross-section for two reasons. First,it is not clear that the mechanical properties of the tibial wall are homogeneousbetween the inter-spur and spur cuticle; and second, it might be purelycoincidental that a spur which may be developed as an anti-predatormechanism also influences mechanical behaviour in an externally evaluated,positive manner. It is possible that the presence of spurs acts to produce stress-88-concentrations that would compromise any improvement in stability derivedfrom their presence. Bertram and Biewener's (1988) model for pre-bend invertebrate long bones was constructed around such a compromise. The bonesare predicted to be pre-bent only when the value of loading predictabilityoutweighs the cost inherent in increased bending loads. Without more detailedknowledge of the consequences of spur formation on the local loading of thecuticle, I am unprepared to commit to an interpretation of the mechanical roleof the spurs in stabilizing the bending deformation of the tibiae.Indeed, the preceding discussion supposes that the spurs are servingprimarily a mechanical role in stabilizing bending of the tibiae in jumping, butit is likely that their design has responded to pressures related to defence frompredation rather than to jumping. Therefore, to fully analyze their design itwould be necessary to balance the design constraints operating on bothbending beams and sharp spines. Because I do not have sufficient quantitativeinformation to fully evaluate these potentially competing design criteria I amunprepared to express an opinion about the mechanical role of the spurs in legbending.CHAPTER 4.HOW HARD DO LOCUSTS JUMP?INTRODUCTION In chapter two the scaling of limb length was shown to produce relativelylonger and slimmer limb segments in larger locusts, an observation that was atvariance with any existing scaling model (McMahon, 1973, 1984, Bertram &Biewener, 1990). It was suggested that the observed scaling relationshipbetween limb shape and body size might represent a design feature associatedwith a jumping mode of locomotion. In order to evaluate this hypothesis weneed to know how the mechanical loading of the limb segments is related tosize across the same ontogenetic sequence.Previous work of other investigators has provided some of this information,but the literature contains conflicting information about the relationshipbetween force production and body size in jumping locusts. Bennet-Clark(1977) showed that the distance travelled as the result of a jump is directlyproportional to the square of the velocity produced in the jump impulse andthat velocity is the result of an acceleration developed over a time interval.However, the accelerations produced in the jump are inversely proportional tothe leg length of the jumping animal. The functional significance of thisrelationship is manifest in the relatively longer legs of jumping animals to reduceloading in the leg skeleton, and the relatively higher accelerations in smalleranimals because of their absolutely shorter legs. For typically observedaccelerations he cites one and a half gravities (g's) for larger vertebrates, suchas leopards and antelope, all the way up to 200 g's for the 0.45 mg rat flea(Bennet-Clark & Lucey, 1967), with mean accelerations of about 24 g's for firstinstar S. gregaria and about 10 g's for adults (Bennet-Clark, 1977). For adultlocusts peak accelerations of about 18 g's are reported (Bennet-Clark, 1975).In contrast, Scott and Hepburn (1976) reported that across theontogenetic sequence of Locusta migratoria there is a relatively constant peakacceleration of about 10 g's produced in the jump impulse. They also reportthat this relationship is demonstrated in a sample of six different species oflocusts and grasshoppers, indicating that this relationship is consistent andgeneral. Further, they observed a consistent relationship between adult,femoral cuticle stiffness and the forces in jumping. They concluded thatconstant acceleration produced in jumping was functionally significant inmatching the changes that occur in cuticular stiffness over the samedevelopmental increase in body mass. Thus, the mechanical properties of thelocomotor structures are appropriately matched to the loads they encounterin normal use.Gabriel (1985a) has reported that over the first four instars, Schistocercagregaria jump approximately the same distance, but the jump distanceincreases by about 300% in adults. Gabriel's data suggest, then, that across thejuvenile instars take-off velocity is similar. If the analysis of Bennet-Clark (1977)is correct, then as the accelerations fall in larger animals, the duration of forceproduction must increase to produce the same take-off velocity (ie. the integralof acceleration over time) across instars. However, if Scoff and Hepburn (1976)are correct in that acceleration is a constant of 10 g's, then the duration of theforce development is also a constant regardless of leg length. These twopredictions would not seem to be compatible, and without a resolution it isdifficult to attempt any functional analysis of the morphological design program-91-described in the previous chapters.It is also important to evaluate the functional significance of the jumpperformance to the animal. Others have suggested that how fast an animalaccelerates is potentially adaptive in that prey that produce high accelerationsare difficult to follow and catch (Emerson, 1978). As a result, the performanceparameter that is thought to be the most functionally significant and hasreceived the closest scrutiny is the acceleration produced in the jump. Emerson(1978) has proposed alternative models that either regulate acceleration or varyacceleration with increasing body size to discriminate strategies for predatoravoidance in frogs. I will make the case that peak acceleration is not thecritical design issue in the jump of the locust.In this chapter I have attempted to quantify the loading experienced bythe legs of the locust during the jump impulse. Using miniature force platetechniques I have measured the force, acceleration, velocity, displacement,kinetic energy and power output during ontogenetic development as well asthe trajectory angle in the jump impulse of locusts on each day of life from thefirst day after emergence from the egg until full sexual maturity (day 45). Thewide range of body mass covered by the locust has allowed me to determinescaling relationships for performance and to compare these relations withexisting models. The data provide clear evidence that the jump plays differentroles in the locomotor performance of the locust at different times its life history.The flightless, juvenile instars are jumping to achieve a functional distance andthe adult locusts are jumping to achieve a velocity critical to the initiation offlight.METHODS and MATERIALS Animal Husbandry. Animals were sampled daily from a breeding colonyof African Desert Locust (Schistocerca gregaria) maintained at the Departmentof Zoology at the University of British Columbia. The animals were kept at aconstant temperature of 27° C, humidity of 56%, and photoperiod of 13:11 (L:D),and fed a diet of head lettuce and bran. A sample of five individuals wascollected each day beginning on the first day following emergence from theegg until approximately two weeks after achieving sexual maturity (ca. 45days).Five jumps from each individual were collected, for a total 25 jumps foreach day in development. Each individual was weighed to the nearest 0.1 mgafter its final jump. Occasionally, the jump event produced un-interpretableforce traces due to transient, large amplitude noise. Video images madeduring several of these jumps indicated that this was the result of slippagebetween the tarsus of the locust and the surface of the platform. Any jumpsthat showed evidence of transient forces were eliminated from the data set.As a result, there are varying sample sizes for each day. Adult locusts wereallowed to jump with their wings intact and unencumbered. Controlexperiments where the wings of adults were held closed with cellophane tapewere not significantly different in terms of peak force production from thosewhere the wings were free to open (t s = 0.067, df. = 14, p<0.05).In this chapter the term impulse is used in the same sense as that inBennet-Clark (1975) in that it does not refer to the exchange of momentumexplicitly, but rather the interval of the jump that can be characterized byground reaction forces above one gravity.-93-Figure 4.1a.Diagram of the force plate used in this study viewed obliquely fromabove. This view demonstrates the relative positions of the windows cut into thehollow box tubing. The arrows show the relative orientation of the forces thateach set of strain gauges perceives; v, vertical forces; h, horizontal forces; f,fore-aff forces. a. position of the force sensing strain gauges.Figure 4.1 b.Diagram of force plate viewed obliquely from below. This view shows therelative positions of the strain gauges (a), balsa wood platform (b), the baseplate to which the force plate was attached (c), the machine screw that heldthe platform in position (d), and the lead wires from the fore-aft gauge (e) andhorizontal gauge (f). The scale bar represents 2 cm.-95-The Force Plate. Figure 4.1 is a diagram of the force plate used in this study.Vertical, horizontal and lateral forces produced in the jump of the locust weremeasured simultaneously with a force plate similar in design to that describedby Full and Tu (1990), but modified to achieve higher sensitivity and frequencyresponse. For instars one through four a balsa wood platform measuring 1.5 x0.75 x 0.15 cm was bonded to the end of a series of hollow, brass box-beams.For fifth instars and adults a larger, circular balsa wood platform measuring 2.5cm in radius and 0.25 cm thick was bonded to the pre-existing platform toprovide a larger surface from which the larger locusts could jump. Within onecentimeter radius of the centre of the platform the position of the locust resultedin a less than 1% change in the force output of the device. Jumps outside ofthis region were eliminated from the data set. The force sensing elements weresimilar to those described by Full & Tu (1990) in that semiconductor strain gauges(type SR4 SBP3-20-35, or SBP3-05-35, BLH Electronics, Canton, Mass.) werebonded to the outer surfaces of double cantilevers produced by machiningwindows in the sides of the box-beams. The mass of the platform was reducedto improve frequency response by having only one set of double cantilevers foreach of the three principal components of force application. The resonantfrequencies ranged from a high of 1.543 KHz for the most distal cantilever pair,to a low of 527 Hz for the most proximal pair. Force sensitivity was enhancedby machining the leaves of the double cantilever down from a thickness of0.015" (3.81x10 -4 m) to a thickness of 0.007" (1.17x10 -4 m). This modificationproduced a sensitivity of 12.60 V/N for the most proximal gauge and a sensitivityof 28.30 V/N for the most distal gauge.Mechanical crosstalk between these gauges was low, but to furtherminimize crosstalk between the vertical and lateral gauges a small piece of-96-telescoping box-tubing was welded into the lumen of the beam between thetwo sensitive elements. Serial calibrations in various orientations in threedimensions indicated that crosstalk interactions where a proximal gaugeinfluenced a distal gauge produced less than a 1% change in the distal gaugeoutput. These were deemed insignificant. Of the three potential interactionsin the other direction only the interaction of the most distal gauge with the mostproximal proved to affect the proximal gauges output by more than 3% and thiswas addressed in the analysis by a recursive loop in the analytical process thatestimated the actual output of the proximal gauge. At no time did noiseexceed 5% of the signal magnitude. Therefore, no signal filtering wasemployed.The resistive elements in the strain gauges formed half of a bridge circuitwhose DC output was amplified and collected at a rate of 5KHz on threechannels of a digital, storage oscilloscope (Data Precision, Data 6000A UniversalWaveform Analyzer, Analogic Co., Peabody, Mass., USA), and transferred to anIBM-PC type computer via serial communications for later analysis.Jumping Arena. All jumps were performed under a clear, plexiglass domeenclosure. The dimensions of the enclosure were 0.45 m wide by 0.33 m deepby 0.58 m tall. The force plate was placed in the centre of the floor of theenclosure on a sheet of Sorbothane, vibration dampening rubber. A balsawood surface measuring 0.25 m by 0.15 m, was constructed with a hole thesame size and shape of the force platform. This surface was placed in theenclosure so that the force plate became a small portion of a larger, relativelycontinuous surface from which the locusts could jump. For each jump event thelocust was placed on the force sensing portion of the wooden surface andallowed to jump freely. Reluctant individuals were enticed to jump with loud-97-noises or abrupt movements in their visual fields. No electrical stimulation wasemployed. The possibility that this enclosure inhibited the performance of thelocusts was not investigated.Data Analysis. The three digitized signals were resolved via vectoraddition into two arrays: one contained the resultant force vector magnitude,and the other an angle to the horizontal for each 200 ps sample. A baseline ofzero Newtons was established by calculating the arithmetic average of the first83 data points (ie. one period of a 60 Hz noise signal) in each force array withthe locust standing on the force plate. Thus, the vertical force records arereported in excess of one body weight. The beginning (or ending) of the jumpimpulse was defined by the force rising above (or falling below) a thresholdlevel that was 2.5% of the maximum force. Force values were normalized bybody mass to produce an array representing the instantaneous acceleration.Each acceleration array was integrated numerically to produce the velocitydeveloped by the centre of gravity for each moment in the jump impulse. Theinitial value of velocity was assumed to be zero, and the velocities wereintegrated to calculate the displacement of the centre of gravity at eachmoment in the jump impulse. Additionally, the product of the ground reactionforce and velocity arrays gave the power produced in the jump. The values ofend-impulse velocity and body mass were used to calculate the kinetic energy(KE) produced in the jump with the familiar formula: KE = lb mass.velocity2 . Thehorizontal distance covered by the jump was estimated using the ballisticequations reported by Bennet-Clark (1981, 1984).Statistics. Except where noted, all statistical tests were chosen based oncriteria presented in Sokal and Rohlf (1981). All relationships were judgedsignificant at the 0.05 probability level. All statistical tests were performed using-98-SO• 70 F-•• 605040 E_5)•^30<^201001.41210.8 E-0.6 E-0.4 E--020.015 -0.01 -0.005 -5040302010Figure 4.2.Sample output for a 0.5007 gram fourth instar individual showing the timecourse of force, velocity, movement and power production. Calculations of thevarious measures of performance are described in more detail in the text.the STATGRAPHICS (STSC, Mass, USA, Ver. 5) statistical software package.RESULTS Figure 4.2 is a representative data set from a single jump of a 0.5007 gfourth instar locust. The force envelope shows the relatively slow developmentof force and the rapid fall in force. The entire impulse lasted 32.8 milliseconds,and peak force was achieved at approximately 27 milliseconds, orapproximately 82% of the impulse duration. In both first instars and adults thetime of peak force shifted later in the impulse duration. This resulted in arelatively more rapid fall in force following the peak force output. Thus, thereare subtle differences in the shape of the acceleration envelope that haveconsequences for the estimation of performance that will be discussed below.This force production profile is not consistent with the optimum described byBennet-Clark (1977) for minimizing the mass of the skeleton. He suggested thatforce should be produced at a constant level during a jump impulse so thatlarge magnitude forces do not produce intolerably high stresses in the skeleton.Ker (1977) has pointed out that this force envelope is the consequence of thecoupling of an unloading, elastic energy storage device with rapidly decayingforce, and a mechanical lever system that is increasing its mechanicaladvantage to take advantage of that decaying force.Within each impulse the angle that the force vector makes with horizontal(ie. the trajectory angle) did not seem to change significantly or systematically.The standard error of the trajectories within a single jump impulse had a highvalue of 1.6 degrees in first instars and a low value of 0.3 degrees in adults.Inspection of the arrays of trajectory angles indicated that the data were notstationary, with the majority of the variation in the data occurring early in the- 1 00-jump impulse when the forces were low and the vector addition of the relativelynoisy force arrays produced relatively high variability of the calculated angles.After the approximate one quarter point in the impulse the forces are higher,making the signal to noise ratio larger, and the variance of the trajectory anglesare even smaller than calculated for the entire time series. This means that it isa fair approximation to assume constant trajectory angle and integrate theacceleration and velocity arrays as scalar rather than vector variables tocalculate velocity and distance moved by the centre of gravity respectively.It also suggests that there is a large degree of stability built in to the design ofthe jumping mechanism of these locusts. The time course of the jump impulse(ca. 15 milliseconds in first instars) makes it unlikely that a neural reflex is capableof modulating muscular control over the jumping mechanism to make fineadjustments to trajectories during the impulse.There is a sexual dimorphism in body mass that develops in the adults,where females become 50% to 75% heavier than males. Analysis of covarianceindicated that for juvenile instars there was no significant effect of sex beyondthe effect of body mass on any of the variables examined (ANCOVA, F s = 3.133,df. = 1, 447, for peak acceleration on body mass, the relationship most closelyapproaching significance.) As such, the data for both sexes were pooled forall juvenile instars for the purposes of performing regressions. In the case of therelationship between movement of the centre of gravity and body mass therewas no significant effect of sex beyond the effect of body mass for the entirelife history (ANCOVA, Fs = 2.737, df. = 1, 874). Therefore, the sexes were pooledfor this entire data set. The distinction between juvenile and adult locusts in thiscontext is not completely arbitrary, as the jump itself may have differentfunctional significance in different stages of the life history (see below).-101-Figure 4.3a.The relationship between body mass to the age of the locust. Data arereported in means and standard errors of the mean in all three parts of thisfigure. A regression fitted to the Von Bertalanfify relation produced the followingdescription of the data: Mass = 3.468(1-e' °59Age)3 (Fs=927.8, df=2,43, r2=.937). Thedata for adults represents both sexes, and as a result, the varience for the adultdata is larger than for preceeding stages.Figure 4.3b.The relationship between measured ground reaction force to the age ofthe locust. A regression fitted to the Von Bertalanffy relation produced thefollowing description of the data: Force = 2.141 (1 -e-.°19 Age)3 (Fs=964.0, df=2,43,r2=.955).Figure 4.3c.The relationship between peak acceleration to the age of the locust.These data are reported as means and standard errors of the means for thepeak acceleration achieved within individual jumps rather than simply dividingthe values in figure 3b by those in figure 3a.Peak Accleration (WO.Ground Rcaction Force (N).^Body Mass (g).a1./FOF.K)^0^0KJ^rt.^•0 0^•-•^151 0■ •Inrr° CDI^I I ^---I ^ I^I^I ^I__ ^I^Both the relationships of body mass and peak ground reaction force asfunctions of age describe sigmoid curves. Body mass increased in a logisticway before levelling off near the end of the fifth instar (Fig. 4.3a.). Fitting a vonBertalanffy growth model to the relationship between body mass and age(Pitcher and Hart, 1982) resulted in a relationship not significantly different fromthat reported in chapter two for a similar series of individuals (F s= 9.187E-5, df.=1,76, p>0.05). Ground reaction force also increases in a logistic manner, althoughmore slowly, and it does not begin to level off until after the moult intoadulthood (Fig. 4.3b.). A similar growth model fit to the data on groundreaction force results in a period of exponential increase in force that is less thanhalf of that for body mass ((e" 057'Age),„ > (e-oirAge)force, Fs=14.808, df.=1, 86,p<0.05). This is manifest in a relatively rapid increase in body mass occurringaround an age of 20 days, while the force increases rapidly around day 30. Thisseven to ten day lag between the age of rapid increase of mass and the ageof rapid increase of force means that normalizing force by body mass producesa NUN-shaped relationship between acceleration and age (Fig. 4.3c.).To analyze allometric relationships log transformations were employed toconvert curvilinear, exponential relationships into linear ones. This has allowedthe convenient comparisons of the scaling exponents that have beentransformed into characteristic slopes. Data from individuals on the first day ofeach instar seemed to be systematically distinct from the rest of the data fromthat instar. It would seem to be the result of the physiological events involvedin moulting that also result in low cuticular stiffness and resilience immediatelyafter moulting (Hepburn and Joffe, 1974, chapters two & three). To prevent thetransient events of cuticular stiffening from biasing the regressions, and to makecomparisons between data collected from individuals that we believe to be-104-functionally similar (chapter one), data from individuals on the first day of eachinstar were excluded in calculating the slopes of the scaling relationships.The relationship between log of peak ground reaction force and log ofbody mass (Fig. 4.4a.) shows a relatively linear region from the smallestindividuals up to the fifth instars, but adults fall above this relation. Theregression of the log of peak force on the log of body mass, including bothjuvenile and adult data, produced the relationship: Log Force = 0.914•Log Mass+ 1.678 (SE = 9.89E-3, r2 = 0.880). The slope of this relationship appears to besimilar to that reported for Locusta migratoria (Scott & Hepburn, 1976).However, an analysis of variance of the residuals showed a significant lack offit (F, = 22.6775, df. = 1, 260, p<0.05), and the regression was deemed aninappropriate description of the data (Draper and Smith, 1981). The regressionof the data consisting entirely of flightless juveniles had a slope of .732 (SE =7.69E-3, r2 = 0.925) and ranges from a low value of 1.99 mN on the first day ofemergence from the egg to a high of 166 mN in fifth instar individuals.The relationship in figure 4.4b. between log of peak acceleration and logof body mass demonstrates an apparent functional relationship over the firstfive instars, but a pronounced increase in performance in adults. The valuesrange from approximately 25 g's in first instars to about 5 to 7 g's in fifth instars.The data at larger sizes, where the accelerations are once again in the 20 to25 g range, are composed entirely of winged adults. As in the relationshipbetween force and mass, a significant, non-random distribution of the residualsin the regression of log of acceleration on log of mass for the entire data set (F s= 22.7327, df. = 1, 260, p<0.05) made this an inappropriate description of thedata. As such, the regression of acceleration on body mass was performedonly on flightless, juvenile individuals. The relationship had a slope of -0.269 (SE-105-Figure 4.4a.The relationship beteen the log of peak ground reaction force and thelog of body mass. Each point is the result of one recorded jump like that shownin figure 2. These data exclude points collected from individuals on the first dayafter each moult. The equation for the regression calculated for juvenileindividuals is Y = 0.992 + 0.732 x X (F s=9045.75, df.=1, 735, r2=.9249). The dashedlines are the 95% confidence limits of the regression line.Figure 4.4b.The relationship between the log of peak acceleration produced in eachjump and the log of body mass. The equation of the regression line calculatedfor juvenile individuals is Y = 0.989 - 0.269 x X (F s=1213.52, df.=1, 735, r2=.6228).The dashed lines are the 95% confidence limits of the regression.Figure 4.4c.The relationship between the log of jump impulse duration and the log ofbody mass. The equation of the regression line calculated for juvenileindividuals is Y = 2.448 + 0.277 x X (F s=4409.27, df.=1, 735, r2=.8571). The dashedlines are the 95% confidence limits of the regression.Log Peak Acceleration (m/s2 ).^Log Ground Reaction Force (N).Log Impulse Duration (ms).^■-•al•....fP.--,.1.0 ^kJKJ. •^s.,^At^NJ•1.7, ^CO^.-•^a.^-J^La4,,‘,. . tO_I^1^L L_ L^1 _1= 7.72E-3, r2 = 0.623).As the animals grow the duration of the force impulse increases in lengthfrom a low value of 12 milliseconds in first instars up to a high value of 65milliseconds in fifth instars, before falling to 20 to 30 milliseconds in adult locusts(Fig. 4.4c.). These data also show a consistent relationship over the first five lifehistory stages before an inflection at the transition to adulthood. The regressionof the log of jump impulse duration on the log of body mass, over the flightlessportion of the life history, had a slope of 0.277 (SE = 4.18E-3, r2 = 0.857), which,while opposite in sign, is not significantly different in magnitude from the slopeof the relationship between acceleration and body mass (t s = 2.913E-8, df. =1,470, p>0.05). This means that as the animals grow through the first five instarsthey produce lower accelerations, but they develop that acceleration over alonger time.By compensating falling accelerations with increasing impulse durations,juvenile instars produce roughly the same take-off velocity in the jumpindependent of age or size (Fig. 4.5.). The flightless, juvenile instars are leavingthe ground at approximately 1.2 to 1.3 meters per second, while the adults areachieving about 2.5 meters per second. The values for adults show goodagreement with the value of 2.63 metres per second based on the jumpdistance of a 3 gram female locust estimated by Bennet-Clark (1975). Forjuveniles, the regression of the log of take-off velocity on the log of body masshas a slope of 0.053 (SE = 5.25E-3, r 2= 0.138) (Fig. 4.5b.). While this slope isstatistically different from zero (ts=10.832, d.f.=736, p<0.05), it only resulted in an21% increase in take-off velocity over the one hundred and seventy fold rangeof body mass covered in the juveniles. In the adults the accelerationsproduced are as high as those produced by the first instars, but the legs are-108-Figure 4.5a.The relationship of velocity produced in the jump to the age of the locust.Data are reported as means and standard errors of the mean. The first day ofeach instar is marked with an *. The dotted horizontal line marks the 2.5 m/sminimum flight speed observed by Weis-Fogh (1956).Figure 4.5bThe relationship between the log of velocity produced in the jump andthe log of body mass. The equation of the regression line calculated for juvenileindividuals is Y = 0.277 + 0.053 x X (F5=117.33, df.=1, 735, r2=.1377). The dashedlines are the 95% confidence limits of the regression line.9tr0 I. O3K 31(CD CDcoop^CI,0 0I^I^I ^d I km▪ ,oo^Witt o arti 9^In ^cl to^0.„ oco I,^II 0^^'00^r linotOno InOn OA-00^190 0 .IA ^^•^^NS<0^CIrn IRV OM CDIIORn up^1fli^ ^ 10 ^ ^ ^14 dill, '" 9onEn no 1^^ 0 00 ^^r$^^^tn0 ^^^Log End-Jump Velocity (m/s).^End-Jump Velocity (m/s).CD^O0approximately five times as long (Chapter two), so both the accelerationdistance and impulse duration are greater than in first instars. As a result, thetake-off velocities are about double those of the previous instars (Fig. 4.5a.).Observation of newly moulted adults indicated that for several day's they do notfly. They spend their time hanging from vertical surfaces basking or flappingtheir wings without leaving the ground. It may be that their wing cuticle requiressome time to dry and harden before it becomes sufficiently stiff to provide anadequate lifting surface. Alternately, the observation that they are flightless forseveral days into the adult stage may reflect the fact that relatively lowvelocities are achieved in the first six days of adulthood compared with thoseafter day 30.- The kinetic energy of the jump, calculated at the end of the jumpimpulse, follows a similar time course as force production, with a large increasein energy in the adult stage (ca. day 30) compared with the juvenile stages (Fig.4.6a.). The values ranged from a low of 0.004 millijoules in a first instar locust toa high of 15.99 millijoules in an adult. The time course of jump energy within thefifth instar follows a parabolic trajectory, where the energy production is similaron the first and last day of the instar. This is similar to that described for the sixthinstar in Schistocerca americana (Queathem, 1991). However, none of theearlier instars show this timecourse. In the fourth instar of S. gregaria the energyrises on the first two days of the instar and then levels off for the remainder. Itmay be that the more rapid development in S. gregaria relative to S.americana (-35 days to sexual maturity vs. -60 days, respectively), and theshorter time spent within each instar masks the changes in the cuticular energytransmission mechanisms to which Queathem attributes the changes inperformance that she observed.Figure 4.6a.The relationship of kinetic energy produced in the jump to the age of thelocust. Data are reported as means and standard errors of the mean. The firstday of each instar is marked with an *.Figure 4.6bThe relationship between the log of kinetic energy produced in the jumpand the log of body mass. The equation of the regression line calculated forjuvenile individuals is Y = 0.253 + 1.114 x X (Fs=11238.60, df.=1, 735, r2=.9386). Thedashed lines are the 95% confidence limits of the regression line.0 10^20^30^40^50I 1,-5 -4.5 -9 -3.5 -3 -2.5 -2Age (days).Log Body Mass (kg).-113-0.0120.002- 1.5-5.50.0 100.0080.0060.009-2.5- 3.5- 9.5oI-The scaling relationship between log kinetic energy and log body mass(Fig.4.6b) had a form similar to that for force and mass (Fig. 4.4a.). Theregression of the relationship for juvenile instars had a slope of 1.114 (SE = 0.011,r2 = 0.939), which is significantly different from a slope of 1.0 (t s = 10.831, d.f.= 735, p <0.05). The adult locusts produced approximately four timesas much kinetic energy as the regression for juveniles would have predicted foranimals of adult body mass.Power developed during the jump follows a time course similar to that forforce production (Fig. 4.7a.). This is not surprising as power is the product offorce, which follows a sigmoid time course, and velocity which is relativelyconstant at the separate juvenile and adult levels. The values for peak poweroutput range from 1.105 mW in first instars to 1.379 W in adults. The values foraverage power output in the jump impulse are about one third of the peakvalues. For the juvenile instars, peak power output scaled to body mass raisedto the 0.772 power (SE = 0.014, r2 = 0.836) (Fig. 4.7b). Values for average poweroutput scaled to body mass raised to the 0.830 power (SE = 0.014, r2 = 0.862).The difference in these slopes is significant (F, = 8.578, d.f. = 1, 1136, p < 0.05),and represents a subtle change in the shape of the force production envelopewith increasing peak acceleration, as mentioned previously.By normalizing the power output of the jump by the amount of jumpingmuscle, we can calculate the specific power output of the jumping musclesand estimate the degree to which elastic energy storage must be employedto amplify the maximal muscular power output of approximately 450 W/Kg ofmuscle (Bennet-Clark, 1975). Gabriel (1985a) has published values for theproportion of body mass that is femoral, jumping muscle in the locust. Shehighlighted a 50% increase in relative muscle mass between the fourth instar-114-Figure 4.7a.The relationship of peak power (o) and average power (0) produced inthe jump to the age of the locust. Data are reported as means and standarderrors of the mean. The first day of each instar is marked with an *.Figure 4.7bThe relationship between the log of peak power (o) and average powerco produced in the jump and the log of body mass. The equation of theregression line calculated for peak power for juvenile individuals is Y = 1.126 +0.772 x X (Fs=2900.77, df.=1, 569, r2=.8360). The equation of the regression linecalculated for average power for juvenile individuals is Y = 0.789 + 0.830 x X(Fs=3546.39, df.=1, 569, r2=.8617). The dashed lines are the 95% confidence limitsof the regression line.ulNr-Iv-IOInOOr-1O•M) intim° Jamod )(um &)-1'OM ) nd) no JamodLog Average Power Output (W).rYOOOU)NOInCBIZ4;•I and adult stages. However, she did not specify the age within the instar thatthese values represent. Additionally, there is quite a lot of variation in the valuesfor relative muscle mass in the juvenile instars, from a high value of 6.1% in firstinstars to a low of 4.3% in the fourth instars, with no clear temporal trend. Forthese reasons I have chosen to average her values for all of the juvenile instarsto produce a value of 5.56% of whole body mass that is jumping muscle for allfive juvenile instars, and use her value of 6.3% for adults.The daily averages for peak, specific power output range from a low of850 W/Kg of muscle in fifth instars to a high value of 5,200 W/Kg in the adultstage (Fig. 4.8a.). However, individual jumps had values as high 11,600 and aslow as 250 Watts per Kilogram. These data agree well with those estimated byGabriel (1985a) based on distance travelled in jumping. It also suggests that onoccasion the power outputs of the jump (250 W/Kg) are well within the rangefor maximal muscle power output (450 W/Kg, Bennet-Clark, 1975). Averagemass specific power output mirrors the data for peak power output at valuesapproximately one third less. If a trend exists, it seems that the specific poweroutput declines from a relatively high value in the first instars down to the fifthinstars before rising to the highest levels in the adults. For the juvenile instars,peak, mass-specific power output scaled to body mass raised to the -0.228power (SE = 0.014, r2 = 0.308) (Fig. 4.8b.). Average mass specific power outputscaled to body mass raised to the -0.170 power (SE = 0.014, r2 = 0.207). Theseslopes were significantly different (F s = 8.578, d.f. = 1, 1136, p <0.05), but similarto the analysis above, I feel this difference is a consequence of the shape ofthe force production envelope.Biewener (1989) has suggested that larger animals may reduce thebending moments applied to their long bones relative to those in small animals-117-Figure 4.8a.The relationship of peak mass specific power (o) and average massspecific power (0) produced in the jump to the age of the locust. Data arereported as means and standard errors of the mean. The first day of each instaris marked with an *.Figure 4.8bThe relationship between the log of peak mass specific power (o) andaverage mass specific power (0) produced in the jump and the log of bodymass. The equation of the regression line calculated for peak mass specificpower for juvenile individuals is Y = 2.417 - 0.228 x X (Fs=253.69, df.=1, 569,r2=.3084). The equation of the regression line calculated for average massspecific power for juvenile individuals is Y = 2.079 - 0.170 x X (F5=148.09, df.=1,569, r2=.2065). The dashed lines are the 95% confidence limits of the regressionline.0lf)0V'OLog Average Mass Specific Power Output (W/kg).(NO^0^00^0^00^0^0co^k0^sr*(2)1/M) ifldill0 .13MOd 3►padS SSEW1^—1I^IVN/M) lndirto Jamod DUPadS s5LW tad Vo'lOu'sOr-IO141by adopting more upright postures. It is possible that large locusts use differentpostures immediately before a jump, and may go through different kinematicmotion during a jump relative to small instars. Such kinematic differences mayaffect conclusions drawn from morphological and mechanical comparisons.Figure 4.9 shows the relationship between the log of centre of gravitymovement during the jump impulse and log body mass. This relationship doesnot seem to show the discontinuity between juveniles and adults seen in theother measures of performance. The regression for the entire data set has aslope of 0.378 (SE = 3.70E-3, r 2 = 0.885), which is not significantly different fromthe slope of 0.375, for the relationship between the log of tibial length and logbody mass for these locusts (Fs= 0.0838, df.= 1, 1363, p < 0.05) (chapter two).This suggests that the distance over which the acceleration is developed is aconstant function of leg length regardless of size or age. It also suggests thatif the jump impulse starts with the knee joint fully flexed, (and this seemsreasonable based on the anatomy of the catch mechanism (Heitler, 1974)),then the jump is kinematically similar in small and large locusts, with no posturalscaling. Additionally, any discontinuity in performance at the transition toadulthood, in terms of velocity or acceleration, is not the result of somekinematic or postural feature of the jump mechanism, but is rather a reflectionof changes in the power generating mechanism.From ballistics we can estimate the distance covered by locusts oncethey leave the ground and become projectiles. Bennet-Clark (1975) presentedthe following formula to predict the distance covered by a ballistic projectile:d = v2 sin26 eq. 4.1.9where d is the distance covered, v is the take-off velocity of the projectile, 6 isthe angle from horizontal and g is the acceleration due to gravity. For a-120-1-5^-4.5^-4^-3.5^-3^-2.5^-2Log of Body Mass (g).Figure 4.9.The relationship between the log of the movement of the locusts' centresof gravity during the jump impulse and the log of body mass. The equation ofthe regression line calculated for juvenile individuals is Y = -0.676 + 0.355 x X(F5=5339.12, df.=1, 735, r2=.8790). The equation of the regression line calculatedfor all individuals is Y = -0.570 + 0.378 x X (Fs=10445.36, df.=1, 1360, r2=.8849).9060706050 30201C;I(o 2C^30Age (days).Figure 4.10.The relationship of the trajectory angle of the jump and the age of thelocust. Data are reported as means and standard errors of the mean. The firstday of each instar is marked with an 41.constant take-off velocity, therefore, projectiles will cover the same distanceover ground regardless of size. In general it is assumed that projectiles arelaunched at 45°, which maximizes distance covered for a given take-offvelocity. Figure 4.10 is a plot of the mean trajectory angle of the jump as afunction of age. These data indicate that locusts are capable of taking off ata wide variety of angles. On average, however, they leave the ground atangles between 45° and 55°, but individuals in this study were observed toadjust their trajectories between 15° and 90° to avoid obstacles in their paths.Given the capability to adjust their trajectory over a wide range, it is difficult toattach much significance to trajectory angles, but in general the trajectorieswere above 45°. Figure 4.11 uses the data on trajectory angle, take-off velocityand equation 4.1 to estimate the jump ranges of locusts on each day that theywere tested. The sensitive dependence of distance on take-off velocity isreflected in the relatively constant distance travelled of 15 to 20 centimetresover the juvenile instars and the three fold increase in jump distance in adults.Bennet-Clark and Adler (1979) have suggested that as much as 10 to 20 % ofthe kinetic energy generated in the jump of the locust may be consumed inaerodynamic drag during the air-borne phase of the jump. Therefore, myestimates of jump distance should be viewed as overestimates, particularly inthe case of the smaller sized locusts.DISCUSSION Scaling of Locomotor Performance. The large data set produced herefor the jump of these functionally and morphologically similar animals that varyin body mass by more than two orders of magnitude allows us to addressquestions of how locomotor performance scales in hopping locusts. How might-123- 10.80.60.4C.2Gr- 11(10^20^2C^40Age (days).Figure 4.11.The relationship of the estimates of the distance covered by the locustsas a ballistic object after they leave the ground and the age of the locust. Adescription of the calculation of estimated distance is in the test. Data arereported as means and standard errors of the mean. The first day of each instaris marked with an M.performance parameters such as force, acceleration or velocity scale withbody mass? Hill (1950) has provided a rationale for relating morphologicaldimensions to force, energy and power output by muscles and I have used thisrationale to generate quantitative scaling predictions. All of the predictedscaling exponents that are generated, as well as the observed values forSchistocerca gregaria are summarized in table 4.1.If muscle force is the ultimate power plant for the locust jump, acceptingthe energy storage role of the apodeme and other connective tissue structures(Bennet-Clark, 1975; Gabriel, 1985b), and the force produced by a muscle is afunction of its cross sectional area (Hill, 1950), then we may be able to predictthe scaling of force if we know how the muscles' dimensions scale. If weconsider a generalized exoskeletal case where the cross-section of the muscleis proportional to the square of the diameter of the limb segment that housesthe muscle, then we can make predictions about the scaling of forceproduction. In the geometric similarity model (GSM) diameter scales to bodymass raised to the 0.333 power (McMahon, 1984). Therefore, force would scaleto mass raised to the 0.667 power (ie. mass 333 • mass 333 = mass 667), andacceleration would scale to mass raised to the -0.333 power (ie.mase67/mass 100° = mass' 333). The elastic similarity model (ESM), which predictsthat diameter will scale to mass raised to the 0.375 power (McMahon, 1973),predicts that force production will scale to mass raised to the 0.750 power, whileacceleration will scale to mass raised to the -0.250 power. Similarly, theconstant stress similarity model (CSSM), which anticipates diameter scaling tomass to the 0.400 power (McMahon, 1984), predicts mass scaling exponents of0.800 and -0.200 for force and acceleration respectively.Interestingly, velocity is predicted to be mass independent in each model.-125-TABLE I. Exponents for allometric equations describing both morphological andperformance measures from models and those observed for Schistocercagregaria. All relationships are modelled by the form x a Mb where M is bodymass in kilograms. The quantity x is the parameter that is scaled as listed below,and b is the scaling exponent.Model Observed forScaled parameter GSM^ESM CSSM S. gregaria.MorphologyLimb length .333t .2501 .200" .377Limb diameter .3331 .3751 .400" .311MechanicsFlexural stiffness 1.333 1.500 1.600 1.532'entire cuticle 1.195Eentire cuticle 0.311Jump PerformanceForce .667th .750 .800 .732Acceleration -.333 -.250 -.200 -.269Velocity .000th .000 .000 .053Energy 1.000t" 1.000 1.000 1.114Power .667th .750 .800 .772Specific Power -.333 -.250 -.200 -.288Movement of centregravityt^McMahon, 1973.333 .250 .200 .378It^McMahon, 1984ttt^Hill, 1950-126-Velocity in jumping animals is determined by the following relationship (Bennet-Clark, 1977, eq. 5):v = ^2s.a Eq. 4.2,where v is take off velocity, s is the acceleration distance (a function of leglength, Above and Fig. 4.9.), and a is the average acceleration produced in thejump. Therefore, we can relate the scaling of velocity to the scaling of leglength and acceleration. For each model, acceleration and leg length havescaling exponents that are equal in magnitude, but opposite in sign. Therefore,their product scales to body mass raised to the zero power and is scaleindependent.Unlike the scaling of velocity, energy is predicted to scale directly withmass raised to the first power, but like the scaling of velocity it is modelindependent. Energy is the product of force, the product of two diameters,and distance, a length. So energy will scale directly with the volume of eitherthe muscle producing the force, or the spring that stored the force.The scaling of power output during the jump can be predicted from thescaling relationships between performance and body mass. Power is theproduct of force and velocity. As we described above, velocity is predicted tobe scale independent by all models. Therefore, power output in the jumpshould scale to mass raised to the same power as does force for each model.So how do the models anticipate the observed relations for Schistocercagregaria? The observed scaling exponent for the dependence of peak forceproduction on body mass (0.732, Fig. 4.4a.) most closely approximates elasticsimilarity; however, it proves to be statistically different from ESM's prediction of0.750 (ts = 2.393, df = 735, 0.1>p>0.05). This scaling exponent for force-127-production in locusts is intermediate to that estimated for vertebrate hoppersfrom anatomical measures. Alexander et al. (1981) reported relationshipsbetween muscle masses, muscle fibre lengths and body mass for a variety ofvertebrate hoppers. In all hind limb muscles reported, the mass of musclescaled very close to body mass raised to the first power. By employingAlexander's (1977) method of estimating cross-sectional area of muscle Iestimated that the fibre areas of deep hind limb flexor muscles of thesevertebrate hoppers scaled to mass raised to the 0.65 power (similar to theprediction of geometric similarity), while in the quadriceps group it scaled tobody mass raised to the 0.75 power (equivalent to elastic similarity), and in theankle extensor group it scaled to body mass raised to the 0.80 power(equivalent to constant stress similarity). This suggests that within themorphological design program that produces geometrically similar amounts ofmuscle, these animals are increasing the muscles' force producing capacity perunit mass. They are presumably accomplishing this by changing the musclefibre architecture, ie. increasing pinnation angle of the more distal limb musclesrelative to those more proximal. It is tempting to suggest that these animals areadopting a distortive allometry (ie. ESM) that increases the cross-sectional areaof muscle in larger animals relative to small ones in a way that biases theincrease toward the distal end of the limb relative to the proximal end. Itaccomplishes this, however, without increasing the relative mass of muscle atthe distal end of the limb, thus preventing an increase in the energetic costs ofaccelerating and decelerating the limbs during locomotion.The scaling exponent of peak acceleration as a function of body mass( -0.269, Fig. 4.4b.) also is most consistent with elastic similarity, but again provesto be different statistically from ESM's prediction of -0.250 Os = 2.471, df. = 735,-128-0.1> p >0.05). The fact that the slope of acceleration on body mass differs fromthe model prediction to the same extent as does the slope of the relationbetween force and mass is not surprising as the acceleration data arenormalized force values. Thus, it seems that the relationships between forceproduction and body mass and acceleration and body mass areapproximating the elastic similarity model. The statistical differences observedmay reflect the influence of morphological scaling that produces longer,relatively more slender legs in larger locusts (chapter two). Nevertheless, asobserved for the mechanical properties of the tibiae, the observedperformance (ie. force production) approximates ESM in spite of amorphological program that deviates rather dramatically from elastic similarity.All models predict mass independence for scaling of velocity in the jump,but we observed that the velocity scaled to mass raised to the 0.053 power.Where does this difference come from? The predictions are based onassumptions about morphology, but data in chapter two show that locust leglengths scale to mass raised to the 0.377 power. Equation 4.2 allows us toeliminate assumptions about morphology and predict how velocity should scalegiven the scaling of leg length observed in S. gregaria. The square root of theproduct of acceleration and leg length (eq. 4.2.) should scale to body massraised to the 0.054 power which is not significantly different from the observedvalue of 0.053. This explains how the scaling of velocity arises, but it does notseem to spotlight a design strategy per se that can explain why locusts producerelatively more elongate legs as they increase in size. That is, we have no apriori reason to anticipate a scaling exponent of 0.05.The scaling slope of energy output that we observe in juvenile instars(1.114, Fig. 4.6b.) is higher than the models predicted. It may be that the-129-relatively larger output of energy seen in larger juveniles is a result of theincreased quality of the material that stores the energy prior to the jump.Before the jump the energy is stored in the extensor apodeme and semilunarprocess (Bennet-Clark, 1975), and during the jump the energy is transmittedthrough the metathoracic tibiae to the ground (Brown, 1963). As such, thecuticular springs form an energy transmission system. Data in chapter twoshowed that the material resilience increases by approximately 20% from first tofifth instars. If we assume that this increase in resilience is reflected in all of thecuticular elements and adjust the amount of energy we measure in the jumpby the energy loss characteristics of the transmission, then we can estimate howmuch energy was input to the system prior to the jump and how this quantityscales to body mass. The estimated energy input to the transmission systemscales with body mass raised to the 1.08 power (F 5=10,129.02, df.=1, 735,r2=0.932), which while closer is still statistically different from the prediction of 1.00made by all the models (t 5=7.9997, df.=735, p<0.05).Whereas each of the other measures of jump performance appear toscale close to the predictions of elastic similarity, peak power output scales tobody mass in a manner most similar to the predictions of constant stress similarity(0.800). It does so in a sensible way, however, as it is the product of force,which scales approximately elastically (0.732), and velocity, which scales tohigher exponent than anticipated (0.053). As a result, power outputs scale tobody mass in a manner that is intermediate between the predictions of elasticand constant stress similarity (0.772). Because power is calculated within eachimpulse, the scaling exponent for power and body mass is numerically differentfrom that which might be predicted simply from taking the product of thescaling relationships for force and velocity (ie. 0.732 + 0.053 = 0.785 vs. 0.772).-130-This difference indicates that the shape of the force and velocity envelopescontribute to the estimate of power produced in the jump. It also suggests thatour examination of the slopes of the energetic relationships are independent;that we are not "boot-strapping" our data to compare the energy or poweroutput and body mass relationships.Overall these results have generated a data set for jump performancethat is in the range of the predictions of models that are based on maintainingscale independance of mechanical parameters. However, the data do notabsolutely match the predictions of any single model. Indeed, force producedin the jump, the parameter whose scaling prediction follows most directly fromHill's (1950) original assertion, scales in a manner closest to elastic similarity whilepower output scales more closely to the predictions of constant stress similarity.Has it been valid, or as suggested in chapter one, naive to doggedly comparethe data on jump performance with the predictions of the various models? Dothe violations of the models' theoretical underpinnings in the scaling ofmorphology fatally compromise their utility?If each scaling exponent that we observe can be explained by the othermeasures of performance, then the data set will represent a single mechanicalpackage--an internally consistent strategy. In the data for locusts there is aconsistent strategy. A route to this conclusion is suggested by the scaling ofmovement of the centre of gravity (Fig 4.9), which, as discussed above, isdirectly reflecting the scaling of leg length. In discussing the scaling of eachmeasure of performance we have seen that there is a rational basis for thestatistical departures away from either similarity model in the context of forceproduction scaling in a manner that is close to, but different from elasticsimilarity and the legs scaling in a manner that produces relatively longer and-131-more slender tibiae in larger locusts. Thus, it is fair to say that the scaling of jumpperformance seen in the locust is following a strategy that is separate fromeither elastic or constant stress similarity, but is itself an internally consistentpackage.Clearly, the original assumptions of both ESM and CSSM are violated inthat leg morphology does not follow either model. Does this violation make itinappropriate to compare the subsequent data on jump performance withthose models? I believe that the comparison is still useful for two reasons.Clearly, we would not know whether the locust was demonstrating anyparticular model unless we actually asked the question. More importantlythough, the results demonstrate that despite the unique morphological scaling,jump performance is scaling in a mechanically functional manner. The fact thatthe data are intermediate to the predictions of ESM and CSSM suggests that themechanical consequences of those strategies (ie. constant strain or constantstress) represent real issues to which the design of the locusts' legs hasresponded. These design issues have had to be dealt with in the context of theobserved morphological scaling, however, and this may have set constraints onthe proximity to which the scaling of jump performance can approach a modelsuch as elastic similarity.I have asserted that elastic similarity is being approximated by the jumpperformance of the locust, but is that reasonable in that no measurementmade so far is statistically elastically similar? My justification for referring toelastic similarity in this context is the scaling of ground reaction force output.Muscle force output is the most independent performance parameter that Imeasured, in that predicting the scaling of force relies on the fewestmorphological parameters. It is also the parameter that scales most closely to-132-elastic similarity. Acknowledging that the locust is not adopting elastic similarityas developed by McMahon (1973) for vertebrate skeletal designs, it isapproximating in performance a design that keeps the normalized deflectionsof a cantilever beam similar independent of size (ie. elastic similarity).Acceleration and Design. Though we have not discussed the data forjump performance in terms of life history strategies, predator avoidance hasbeen the context for previous analyses of jumping in locusts and anurans (Scottand Hepburn, 1976; Emerson, 1978; and Queathem, 1991). In each case thesuggestion has been made that acceleration produced in the jump is thecritical performance parameter. In discussing jumping in anurans, Emerson(1978) used scale independence of jump performance parameters todiscriminate models that associated specific measures of performance withsuccess in avoiding predation. She found that in Rana pipiens and Pseudacristriseriata average acceleration was relatively scale independent over a 30 foldrange in body mass, while Bufo americanus showed a decrease in averageacceleration with increasing size. By this criteria the ontogenetic data onlocusts suggest that acceleration per se is not necessarily the key functionalperformance feature, but rather the velocity developed in the jump is perhapsmore important. This suggestion is based on two observations. First, velocity,which is relatively scale independent over the first five instars, rather thanacceleration, which varies four fold, is regulated by the developmental designprogramme. While there seems to be a functional relationship between fallingaccelerations and increasing body mass for juveniles, the duration of forcedevelopment in the jump seems to be compensating quantitatively for the fallin acceleration to provide a relatively constant take off velocity. The scale-133-independence of take-off velocity suggests that if selection has operated onthe jump performance of this locust, then velocity, or a consequence ofvelocity, is the key performance parameter. Secondly, there seems to be anunexploited potential to improve acceleration performance in flightless instarsthat is exploited to some extent in adults.Figure 4.12 provides a comparison of the accelerations produced injumping locusts with published values for the flea, Spylopsyllus cuniculus (Bennet-Clark and Lucey, 1967), click beetle (Evans, 1971), the mediterranean fruit flylarvae (Maitland, 1992), and the standing jump of the Kangaroo rat (Bieweneret al., 1988). If we perform a two-point regression between a value for theaverage male adult locust and the flea (of all the animals reported in figure4.12 these the flea and the locust are the ones that we know jump using similarcuticular spring mechanisms), the slope turns out to be -0.2499. The slope of therelationship between acceleration and mass predicted by ESM of -0.25 wouldseem to describe the adult fleas and locusts even better than the slope of -0.269 which describes the flightless locusts. The fruitfly maggot and kangaroorat both produce accelerations that lie very close to the regression based onthe performance of the flea and the locust. This suggests that the samefunctional design issues that determine scaling relationships in flightless, juvenilelocusts act to determine the separate relationships for adult locusts and theseother jumping animals of widely different designs. Apparently the robustpredictions of ESM apply to these jumping animals generally, but the juvenilelocusts are able to get by (in an evolutionary sense) with about one third loweraccelerations per body mass than the heroic performance of the adults andthese other animals.A feature of these relationships is that if elastically similar jumping animals-134-Log of Body Mass().Figure 4.12.The relationship between the log of peak acceleration produced injumping and the log of body mass. This figure compares the data reported infigure 4c with published values from other jumping animals. Data is presentedfor the flea (o), Spylopsyllus cuniculus, from Bennet-Clark & Lucey (1967), theclick beetle (•) from Evans (1971), the mediterranean fruit-fly larvae (I•) fromMaitland (1992), as well as the standing jump of the kangaroo rat (•) fromBiewener et al. (1988) and the continuous hopping of the kangaroo (•) fromAlexander & Vernon (1975). The equation of the two-point regression madebetween the data from the flea and average calculated for all of the adult,male locusts had a slope of -0.2499, which seems indistinguishable from theprediction of elastic similarity (-0.250).-135-increase in size and travel down the line relating acceleration with body mass,there is a point where the accelerations fall to a level where the animal willbeunable to leave the ground in a jump, and the body mass where this occurswill be an absolute limit to body size for a given jumping 'design'. Realistically,a functional limit will occur before this point, as decreasing accelerationsproduce slower and shorter jumps in the absence of spectacular compensatoryspecializations of leg length to increase the interval over which acceleration isdeveloped. In any event, it seems that an animal that is elastically similar tojuvenile locusts could not continue to get larger indefinitely, and for largeranimals to jump they must move off the relationship between acceleration andbody mass for juvenile locusts and produce higher accelerations at a given size.There seems to be two design strategies that could allow an animal tomove off of any of these relationships; either produce more force per unit bodymass (ie. increase the proportion of body mass that is jumping muscle), orincrease the efficiency of the energy transmission system in producing groundreaction force. Gabriel's (1985a) data on locusts suggest that adult locustshave increased the proportion of body mass that is jumping muscle to achievethe high accelerations seen in the adults. Gabriel (1984) made the pointpreviously that in larger animals, where energy production capacity limitsjumping performance, increasing the relative investment in jumping muscle isthe most effective way of improving jumping performance. Clearly 40 kg.Kangaroos are not 'elastically' similar to adult locusts (Fig. 4.12.), and it has beenobserved that as much as 8 to 10 % of the body mass is invested in hind legmuscle of a kangaroo (Alexander & Vernon, 1975) compared to 4.5 to 6 % inlocusts (Gabriel, 1985a). Also, kangaroos are storing kinetic energy from onestride and using it in a following stride while locusts use only energy that is stored-136-during a single 'stride'.It seems significant that the data for standing jumps in kangaroo rats andmediterranean fruit fly larvae lie so close to predictions based on adult locustsand fleas. In both cases the animals use quite different jumping mechanismsfrom the locust. Indeed, the fruit fly maggot is legless. In continuous locomotionthe Kangaroo rat is known to use elastic energy stored from previous strides toincrease the accelerations and energy output of a jump (Biewener et al., 1981),but in single jumps from standing starts they are not known to use stored springenergy. It is therefore interesting to see the data for a standing jump in theKangaroo rat so close to the prediction made by adult locusts and fleas (Fig.4.12.). This relation may reflect common design features in very differentanimals. It could be that for a given relative investment in jumping muscleeach animal gets a similar acceleration output independent of morphologicaldesign. The limits to this suggestion are demonstrated by both the click beetle,which has a similar investment in muscle as the locust, but much higher relativeaccelerations (3800 mist , Evans, 1971), and the fruit fly maggot, which hassimilar accelerations to the locust, but much larger investment in jumpingmuscle (16%, Maitland, 1992). It remains to be discovered if this relationship,which suggests that this variety of jumping designs are elastically similar, is otherthan a coincidence.In going from the line for juvenile locusts in fig. 4.12 to the line for adultsthe relative muscle mass increases approximately 20%, but the kinetic energyoutput goes up by four fold (Fig. 4.6b.). Does the 20% increase in relativemuscle mass between fourth instars and adults explain the increase inperformance observed at the transition to adulthood? Where might additionalenergy come from? Gabriel (1985b) made the observation that the distance-137-covered as a ballistic projectile increased by 300% between fourth instarhoppers and adults. She explained that the increased energy produced in thejumps of adult locusts was the product of relatively stronger muscles applyingmore force to stiffer energy storage devices, and therefore storing more energy.She noted that adult extensor muscles' pinnation angle increased and relativemuscle fibre length decreased relative to fourth instar locusts, indicating that themuscle was capable of producing more force per unit volume. She alsoshowed that the adult apodeme increased in cross-sectional area by 440%,presumably increasing in stiffness relative to the fourth instar hoppers, and thesemilunar process increased in measured stiffness by six fold (Gabriel, 1985b).She felt that the increased stiffness of the spring combined with the increasedforce output of the muscles could account for a large part of the increase inspecific energy seen in the jumps of adults. This analysis, however, does notseem to describe the situation in locusts adequately. Gabriel reports functionalcross section of the tibia extensor muscle as 5.7 mm 2 and 18.9 mm2 for the fourthinstar and adult respectively. When the area is normalized by the body massbeing accelerated, the values are the same for the different age classes (20.40mm2/gate wear vs. 20.58 mm2/gAduits). Similar forces applied to more easilydeformed springs could also store relatively larger amounts of energy, but thisalso does not appear to be the case. Gabriel's data show that the forceproducing cross-sectional area of muscle is applied to a relatively larger cross-sectional area of apodeme in the adults compared to the fourth instars (1781mm2 muscle/mm 2 apodeme4th ,„tar vs. 1092 mm 2 muscle/mm 2 apodemeAdults).This results in the apodeme springs seeing 37% less force per unit spring area inadults. What we have, therefore, are similar forces applied to relatively stiffersprings resulting in smaller deformations of the springs and smaller amounts of-138-energy stored in adults relative to fourth instars. We are still left wonderingwhere the additional energy comes from in the adults' jumps. It seems verylikely that Gabriel's suggestion that the juvenile muscles are not working as hardas they are in adults is correct (Gabriel, 1985; Gabriel & Sainsbury, 1982).The Ontogenetic role of Jump Performance. Bennet-Clark (1977) showedhow body size plays an important role in jumping design. Chiefly, this results inthe jumps of small animals being limited by their ability to generate poweroutput, while the jumps of larger animals are limited by their ability to generatekinetic energy. Gabriel (1984) analyzed these ideas and predicted that small,power limited animals can achieve better performance by either getting largeror increasing their leg length. In the largest jumpers she felt that improvementsin performance could optimally be achieved by increasing the fraction of thebody mass that was committed to jumping muscle. Indeed, she reported thatacross the ontogenetic increase in body mass in locusts, jump performanceimproved by increasing body size up to the size of fifth instars. In adults,however, jump performance was improved by increasing the relative mass ofjumping muscle by approximately 50% over that in fourth instars (Gabriel,1985b).These results indicate that the increase in jump performance seen inadults does not represent a change associated with body size, but rather achange in life style. It is no mere coincidence that the increase in jumpingmuscle occurs at a point where the mode of locomotion switches from hoppingto flying. I suggest that one strategy may not appropriately describe thedevelopment of the jump performance. Rather it may be more reasonable todevelop one framework to analyze the juveniles, and another to analyze the-139-adults; frameworks that are constructed to account for what may be entirelydifferent ecological roles of the jump at different points in the life history. Insmall locusts we see an increase in leg length that is rapid relative to theincrease in body size (chapter two) as a strategy for increasing jumpperformance. The increase in jumping performance in adults reflects a switchto flight, and the demand that flight makes for higher performance jumps.Indeed, this distinction has already been alluded to in discussing locust jumpperformance data (Scott & Hepburn, 1976; Gabriel, 1983; Queathem, 1991). Thisinterpretation seems fundamentally separate from the switch to largerinvestment in jumping muscle mass in larger jumping animals that Gabriel (1983)anticipates. If I was keen to shoe-horn my observations into Gabriel's paradigm,then I might say that the locust exploits the increasing leg length strategy up tothe point where the locomotor strategy changes and there is a demand for afundamentally different, and stronger jump. If this represents a strategy, thenI view it as a locomotion mode change issue, rather than a body size issue perse.The ontogenetic increase in body mass and increase in peak forceproduction are the result of a clear difference in developmental timing. Therelatively early period of rapid mass increase would seem to indicate that foryoung, small locusts getting larger is a higher priority than increasing forceproduction to maintain high accelerations in the juvenile instars. Once thelocusts have achieved a large adult mass, the delayed period of rapid increasein force production results in relatively high forces that produce largeaccelerations quantitatively similar to those observed in first instar individuals.Acceleration is dependant on the amount of force produced by thejumping muscles and the body mass being accelerated. So the adults are-140-producing 25 g's of acceleration by producing more force per unit body massthan the flightless fifth instars of similar body mass (Fig. 4.12.). If survivorshiprequired as high acceleration performance as possible, then why wait untiladulthood to increase the force output of the jumping machinery?Acknowledging the teleological danger in asking 'why" type evolutionaryquestions, it still seems that the high acceleration per body mass seen in adultscould be achieved in juveniles with benefits to predator avoidance byadvancing the developmental timing of jumping muscle growth. Certainly ithas not been necessary to invoke predator avoidance as a 'strategy' for thejump in the flea. Therefore, having high peak-acceleration as a strategy forpredator avoidance seems questionable. If the jump of the locust producesone velocity, and therefore distance, in flightless individuals, and anothervelocity in winged ones, what then is significant about these two levels ofperformance? In flightless, juvenile instars the jump data predict a distancetravelled of approximately 20 cm, a figure that is intermediate between themaximum performance of 30 cm and average of 11 to 14 cm for fourth andfifth instars observed by Gabriel (1985a). Regardless of the specific distance,the interpretation is the same: there appears to be a functional distance thatis important to the hopping locust. To test this hypothesis it would be importantto look for some characteristic dimension in the locust's environment thatcorrelates with the jump distance. It may be that there is a pattern to thespacial organization of the vegetation that forms the food source for thehoppers that has a characteristic spacing of 20 to 30 centimetres.In adults the important parameter may not be distance travelled, butrather take off velocity itself. It is significant that the two fold increase in takeoff velocity occurs at the point in the life history where the primary mode of-141-locomotion switches from hopping to flying. Weis-Fogh (1956) reported thatlocusts flying in wind tunnel experiments stop flying when the wind speed fallsbelow two and a half meters per second. It seems unlikely that the minimumobserved flight speed and the take off velocity in adults being the same issimply a coincidence (Fig. 4.5a). Because locusts use unsteady-stateaerodynamics during take-off it is difficult to estimate what their minimum flightspeed might be a priori. However, it seems reasonable to estimate what effectthe increased take-off velocity might have on thrust production using actuatordisc theory. With actuator disc theory we need not be specific about theanatomy or the mechanics of the thrust generating mechanism. For actuator-discs the thrust produced is proportional to the mass flux of fluid moving throughthe actuator times the velocity of the fluid. The quantitative relationship isT = p Sd v (V + 1/2 V), Eq. 4.3.where T is the thrust produced by the actuator-disc, p is the density of the fluid,Sd is the disc area--calculated as a circular disc swept out by wings of a knownlength, V is the forward velocity of the actuator relative to the fluid and v is theincrement of velocity added to the fluid as it passes through the actuator disc(Blake, 1980). Figure 4.13 is a plot showing real solutions of equation 4.3 for v interms of varying values of V. The data in figure 4.13 suggest that as theactuator disc moves through the fluid at higher velocities (V), the increment ofadditional velocity (v) falls rather quickly. For a 3.2 g female locust with 5.5 cmlong wings to take off from a standing start, the wings must generate an air-flowof 2.62 m/s to generate sufficient thrust to balance body weight. By jumping offthe ground at 2.5 m/s at an angle of 55° to horizontal, the additional velocitythat the wings must generate falls to 1.12 m/s - a decrease of 57%. Were theadults to reach the same end-jump velocity as the fifth instars the wings would-142-Figure 4.13.Increment of additional velocity required to produce thrust at 55° (ie.trajectory angle) large enough that its downwardly resolved component issufficient to balance body weight for actuator discs moving at variousvelocities. The data are solutions for the following equation relating thrustproduced by an actuator disc to the velocities of the fluid moving through thedisc: Thrust = p Sd v (V + 1/2 v), where p is the density of air, Sd is the area of thedisc (the area swept out by wings of given length), V is the velocity of the disc(modelled here as the end jump velocity of the locust), and v is the incrementof additional velocity to which the fluid must be accelerated as it passesthrough the disc in order to generate the required thrust. This formulationassumes that the actuator disc is oriented normal to the trajectory angle. Abody mass of 3.2 g, a wing length of 5.5 cm and a density of air of 1.1746Kg/mm 3 were used to calculate the data labelled 'Female' in the figure. Amass of 2.0 g and a wing length of 4.8 cm was used to calculate the datalabelled 'Male' in the figure. The data indicate that producing 2.5 m/s in thejump of a 3.2 g female requires the production of 35.6% less additional velocitygoing through the disc, and 73.3% less power required, relative to an end-jumpvelocity of 1.1 m/s. The data also indicate that jumping at 2.5 m/s requires57.2% less additional velocity than a standing start with a power requirementsavings of 92%. The estimates for the smaller, male locusts suggest that theincrease to adult take-off velocities lower the additional velocity required by37.5%, and reduces the power required for take-off by 75.6% over therequirements imposed by the juvenile end-jump velocities.-143-2.5-Male (2.0 g)Female (3.2 g)0.5-0 I.5^1^1.5^2 2.5 3^3.5 4 4.5Disc Velocity (m/s)have to generate a flow through the disc of 1.74 m/s, a decrease of only 33%over a standing start, and 55% greater than the increment of velocity for a jumpof 2.5 m/s. For a 2 gram male with 4.8 centimetre long wings the data in figure4.13 suggest that jumping with a velocity of 2.5 m/s requires an additionalincrement of velocity of 0.95 m/s, a reduction of 60% from the standing-start discvelocity of 2.37 m/s. Since the power required by an actuator is proportionalto the third power of the disc velocity (Von Mises, 1959), the difference in end-jump velocities between fifth instars and adults represents a 75.6% savings inpower required to generate the thrust necessary to overcome the force ofgravity. This power savings seems considerable, but without knowing explicitlywhat the power requirements are for flying at 1.1 m/s relative to the maximalpower output capacity of the flight muscle it is impossible to say if the jumpingvelocity developed by juveniles represents an absolute limit to flight by locusts.Alternately, it may be that to jump much faster requires uneconomically highjumping muscle power output and the observed jumping velocity represents abalance between the falling demands for power from the flight muscle and theincreasing demands for power from the jumping mechanism with increasingend-jump velocity.These observations do suggest that achieving a high take-off velocity forthe initiation of flight has demanded a higher performance jump than could beprovided by the force production of the juveniles, and I believe that thisincreased demand for performance is the design issue that has driven theincrease in power output in the jumps of adults over those of juveniles.CHAPTER 5.GENERAL DISCUSSIONIn the case of each measure of mechanical performance that we candirectly relate to morphological predictions (ie. flexural stiffness, force andacceleration) elastic similarity is approximated. In the case of power outputand specific power output, which scale in a manner that is closer to constantstress similarity, the observed scaling for S. gregaria is quantitatively reasonablebased on the dependence of power output on the other performancecharacteristics. However, the approximation of elastic similarity is achieved inspite of a morphological design program that produces increasingly spindlylegs, and so deviates dramatically from elastic similarity.If we were to adopt the adaptationist's paradigm, we might find theapparent approximation of elastic similarity a satisfying result. If function in acantilever structure is dependant on controlling the deformation in bending,then as outlined in chapter one, if we want to adopt a similarity that results indevices to experience the same normalized deformation. This is the foundationof elastic similarity, and it provides a rationale for the locust's implementationof ESM in the mechanical design of its legs.So locusts appear elastically similar in terms of mechanics, and we thinkthat for bending structures this is a reasonable strategy. But if there is somevalue in adopting elastic similarity in the mechanical design of the legs, thenwhy not be elastically similar in morphology as well? I can provide threesomewhat speculative arguments to explain the morphological design that is-146-observed in the locust. Each is formulated in a separate context, but no singleargument is exclusive of the others.The manufacturing issue. It is possible that for a given adultmorphology the constraints imposed by the process of moulting in exoskeletalanimals determines the dimensions that the developing locust may adopt inproceeding toward the adult dimensions. Examination of the data in figure 2.5& 2.6 suggests this possibility. Between each juvenile instar the externaldimensions increase by a similar percentage. The lengths of the tibiae increaseby approximately 50% while the diameters increase by approximately 40% aftereach moult. The process of moulting involves drawing the soft cuticle of thesubsequent instar from within the hardened exuviae of the previous instar.Given the dimensions of the lumen of the old exoskeleton, it may be that thenew cuticle's dimensions can only be increased up to a maximal amountthrough this drawing process. The scaling of the external dimensions is,therefore, indicative of the time spent within each instar and the amount thatbody mass increases within the same life history stage. Thus, it may or may notbe demonstrating a specific design strategy that attempts to produce goodjumpers throughout the life history of the insect.The moult from the fifth instar to adult results in a smaller increase indimension than any previous moult, suggesting that perhaps the moultingprocess does not limit the path that the external dimension may take in arrivingat the adult condition. There are a number of distinctive changes that occurat this one moult, and it may not be fair to compare the process with theprevious moults. For example, we have observed in chapter four that themechanical events in the jump show sharp discontinuities at the moult to-147-adulthood. Therefore, the musculo-skeletal reorganization involved in thistransition may place further constraints on the potential for change in theexternal dimensions occurring within a moulting event. Indeed, it is significantthat there is a discontinuity in the scaling of each of the jump performanceparameters at the moult to adulthood, but no discontinuity is observed in thescaling of dimensions or material and structural properties. A possibleinterpretation is that if the locust needs to be a competent hopper as an adult,and this requires a suite of parameters to have particular values, then growthin dimensions of the skeleton must follow a specific trajectory in time. Economy,however, does not likewise require performance parameters to follow a uniquetrajectory. So the juveniles can survive by producing lower forces per unit bodymass than adults, resulting in the discontinuities seen in the moult to adulthood.However, the fact that the average body mass increases from fifth instars toadults in a way that maintains the scaling relationship between externaldimensions observed across the juvenile instars suggests that there is a specificdesign program that is determining the scaling of the morphology. Thefollowing arguments discuss the nature of that specific strategy.Force vs. body mass scaling. The morphological predictions that werelaid out in chapter one were based on the loads experienced by the limbsbeing related directly to body mass. In chapter four we saw that the peakforces produced in jumping locust legs are five to twenty times body mass andscale to body mass raised to the 0.732 power. If the scaling programme isresponding to peak ground reaction force rather than body mass, then it maybe inappropriate to expect morphology to scale in an elastically similar manner.We could ask if force is not increasing as fast as the models predicted, then-148-what are the consequences of the observed scaling of external dimensions ofthe locust's legs?If we re-examine equation 1.1a = (Fly) ^Eq. 1.1.and use the observed scaling of peak ground reaction force, limb length,diameter, and second moment of area, we generate the following relationshipfor the scaling of peak stress:a oc (mase.732) (mass0.377 (mass0.311) = mass0.218(Mass I2 )and remembering that strain (e) is a/E,e (Mass°732) (Massa3771 (Mass0.311) = Mass-a 112(Mass L5 2)Eq. 5.1,Eq. 5.2.Because the uncertainty of each scaling relation contributes to the uncertaintyof the overall scaling of stress and strain it is impossible to distinguish the slopeof the scaling of strain from a value of zero. Therefore, we cannot reject theidea that the leg's dimensions are actually adjusted to maintain a functionalelastic similarity that is responding to the scaling of loading, rather than to thescaling of body mass per se.Unfortunately, if we insert the predicted morphological scaling exponentsinto the above relationship where force scales to mass raised to the 0.75 power(the prediction from chapter four), we anticipate that strain will scale to bodymass raised to the -0.125 power. This is also statistically indistinguishable fromthe exponent generated in equation 5.2. It seems that in order for the legs toactually follow a functional elastic similarity in the face of peak loads that arenot increasing as fast is body mass, the locust would require even longer and-149-more spindly legs than it actually has. In fact, to maintain volume in such legsthe length would have to scale to body mass raised to the 0.5 power while thediameter scale to the 0.25 power, producing legs that scaled length to thediameter raised to the 2.0 power.It is important at this point to remember that this analysis is predicated onthe assumption that muscle force scales to the 0.75 power of body mass. Thisassumption is based on Hill's (1950) suggestion that muscle force is proportionalto a functional muscle cross-sectional area which in turn is the product ofdiameter squared. If we are going to suggest that the dimensional scaling oflimbs loaded with forces that scale to massam should produce even greaterspindliness, then the prediction of 0.75 will change as well. In chapter four weassumed that the muscle in question is parallel fibred as a simplification, but infact many muscles, the jumping muscle of the locust included, are highlypinnate. As such, our functional connection between the dimensions of themuscle and the functional cross-sectional area need to be adjusted. The fibresin a pinnate, jumping muscle will insert on the femoral wall and the apodemetendon. The area of this insertion will be proportional to the cross-section of themuscle and will be proportional to the product of the length and the diameterof the insertion, rather than the square of the diameter of the muscle-housinglimb segment.If we once again reflect on equation 1.1, but this time resolve eachmechanical parameter as a scaled function of dimensions and deal withpinnate muscle, the relationship between dimensions and strain becomes(i • d)(d)(0 = Constant(d)4Remembering that a constraint of constant volume demands M = 1. d2 leads-150-to both length and diameter scaling to body mass raised to the 0.333 power--geometric similarity. An important part of this prediction is that force will scaleto body mass raised to the 0.67 power, not 0.75, and I will scale to mass raisedto the 1.33 power, not 1.5. Therefore, in situations where the forces that thescaling strategy are responding to are produced by muscles I would anticipatethe predictions of geometric similarity and elastic similarity to beindistinguishable. It is perhaps not surprising, therefore, that Alexander et al.(1979) found such a wide spread expression of geometric similarity in a widevariety of animals.These calculations indicate that for animals where the peak loads arereally functions of body mass, elastic similarity will produce limbs that becomeincreasingly stout as McMahon anticipated (1973) and observed (1975).However, when the peak loads are functions of muscle forces, the limbs shouldproduce geometric similarity. It is tempting to suggest that this differencerepresents a real difference in animal design. Indeed, any prediction that forcewill scale to body mass raised to a power less than one suggests that "spindly'elastic or geometric scaling will occur. In small animals, where the absolutevalue of the muscle forces are high relative to body weight, I anticipate this kindof strategy. In large animals, where the force of gravity will be large relative tothe forces generated by the muscles, I would expect to see 'stout' elasticsimilarity.This discussion suggests that Prange's (1977) observations thatcockroaches and spiders are apparently geometrically similar could also beinterpreted as observations that these invertebrates are elastically similar.Locusts, however, have large pennate jumping muscles and are notgeometrically similar. The scaling of force output does not follow from the-151-scaling of the product of a length and a diameter (0.732 for„ vs. 0.3- 77Iength +0.3 11 dia ter = 0.688). What does this mean? There are two parts to the answer:how and why. With respect to how, it is likely that the diameter of the femorascales to body mass in a manner different from the tibiae. I would predict thatfemoral diameters scale to mass raised to the 0.355 power, which would stillproduce spindly morphology, but would produce a predicted scaling of forceto mass raised to the 0.732 power. The discussion of which force the designstrategy is responding to has also ignored the effect of changing the materialproperties of the skeleton. Given a mutable material stiffness, an additionaldegree of freedom in adapting the structural design of the skeleton to thechanging demands of increasing body size is supplied, and the limits to exoticmorphologies are only limited by the scope of the material to stiffen. Withrespect to why, I would suggest that there is either a developmental constraintas outlined above as a manufacturing issue, or there is a functional role for thespindly morphology that the legs express that is associated with the jumpingmode of locomotion.Spindly levers as design strategies. If a design strategy could accomplishelastic similarity with a traditional model, where the limb skeletons becomeincreasingly stout with increasing size, why then adopt a spindly strategy thatproduces fundamentally larger deformations, risking catastrophic rupture? Doesthe spindly strategy provide some benefit in performance? An answer isprovided by turning the question around and asking what is the difference inperformance if the tibia is relatively rigid or has some degree of compliance?-152-Figure 5.1a.Diagramatic expression of how falling spring energy (SE) is balanced byincreasing mechanical advantage (MA) to produce a parabolic groundreaction force (GF) envelope.Figure 5.1 b.Diagramatic expression of the role of the energy time machine' inherentin the locust legs design. In unloading stored spring energy into a rigid lever theforce pays off against mechanical advantage to produce ground reactionforce that follows the same parabolic trajectory as in figure 5.1 a (doffed line).If some of the energy from the first spring is put into deforming a second spring,which pays back that energy in later when mechanical advantage is higher(heavy solid line), there is a delay in the point where peak ground reactionforce is achieved. The consequences of this change are discussed in the text. Spring EnergyMechanical AdvantageSpring EnergyMechanical AdvantageVDCD=-!CDeon] 4ndro ao.ioj 4nd4noAs mentioned in chapter four, Ker (1970) has pointed out that theproduction of ground reaction force during the jump of the locust is the productof the force stored in the spring, which decays gradually as the spring energyis converted to kinetic energy of the accelerated mass, and the mechanicaladvantage, which increases in an approximately inverse manner to the decayof spring force (Ker, 1970; Bennet-Clark, 1973 for locusts). Figure 5.1 is a diagramrepresenting this trade off of spring energy and mechanical advantage for atheoretical spring system. If the lever arm whose mechanical advantage ischanging is itself compliant (in the sense of being deformable rather thanspecifically 1 /E), then it is possible that some of the spring force will go intodeforming the lever which would then be returned later in the impulse at a timewhen mechanical advantage is high. Thus a spatially intermediate springcould provide a sort of 'time machine', delaying the decay of spring energyuntil the mechanical advantage is able to make better use of the spring force.The dotted line in figure 5.1 b represents the consequent trajectory of the springenergy decay of this design model. The heavy solid line is the resultant groundreaction force (ie. the product of the spring force and the mechanicaladvantage). If this model works, then the spindly legs of the locust may actuallybe designed to take advantage of the inherent deformability of long slenderbeams loaded in bending.I have tested this design idea with a simple mechanical model shown infigure 5.2. A quarter inch steel rod was placed on a pivot and suspended witha Pesola spring scale. A 500 g mass was suspended from the rod and the rodwas then depressed, preloading the spring scale to 1.40 kg and held in placeby a piece of twine. The twine was set aflame and allowed to burn through,allowing the spring scale to unload and accelerate the mass. The entire-155-Figure 5.2.Diagram of the mechanical model used to test the idea that compliantlevers have a functional role in delaying the point of peak force. The deviceconsisted of a rigid lever (a) mounted on a pivot with a teflon bushing (b). APesola spring scale was mounted on the lever (c) and was initially loaded to 1.4Kg beyond the 0.5 kg mass being accelerated (d). The lever was held in thepreloaded condition with a piece of string (e) attached to a concreate weight(below this view of the device). The experiment consisted of lighting the stringon fire and observing the entire sequence of events on video. When the stringbroke and the mass was accelerated, its movement was noted on a piece ofacetate film placed over the video monitor. For the experiment that modelleda compliant lever, a rubber band was placed between the mass and itsattachment to the lever at point f.-157-experiment was monitored on video, and the position of the mass was observedby advancing the video tape one field at a time, and noted on a piece ofacetate film placed over the video monitor. The velocity and acceleration ofthe mass were estimated by differencing the position and velocity datarespectively. The entire experiment was then repeated with a rubber bandplaced between the mass and the rod to model a degree of flexibility in theenergy transmission system, similar to the tibial flexability of the locust leg.In modelling the locust leg it seemed appropriate to define a referencepoint to evaluate the performance of the model. I have evaluated the modelby comparing the velocity achieved by the mass at a point in time whichrepresents the feet leaving the ground. What does that mean for a rotating rodand mass? I decided, perhaps arbitrarily, to define this "takeoff' point as thedisplacement where the mechanical advantage reaches a maximum, as seemsto be the case in the locust. Therefore, the experiment was set up so that thespring scale gave 0 force, beyond the weight of the mass, when the rod washorizontal. The range of motion of the spring scale limited the prejumpextension to 20 cm, and unfortunately, this really did not allow for a realisticallylarge range of mechanical advantage as seen in the locust, but the pertinentfeatures of the model are retained. For comparison purposes the velocity of themass when it had travelled 20 cm were evaluated.Figure 5.3 is a plot of the time course of the displacements of the massconnected rigidly to the rod and for the 'compliant' rod. These data indicatethat with the rigid connection the mass reaches 20 cm of displacement atapproximately 180 ms; whereas with the compliant linkage, the mass reaches20 cm at approximately 215 ms. Thus, the rigid lever has a higher averagevelocity over that interval. However, the position of the compliant lever was-158-. ......... .Compliant LeverRigid Lever0.30.25-0.2-EE?E 0.15-0.1-0.05-50^100^1 50^200Time (sec)250^300Figure 5.3.Plot of the displacements of the mass in figure 5.2 through time for therigid lever (dotted line) and the compliant lever (solid line).-159-••.21.8-1.6-1.4-i 1.2-Z% 1-8ai> 0.8-0.6-0.4-0.2-o...: %•Rigid Lever. .//•••a••••. \^:^ *•., Compliant Lever. S. ;...a;.. S.....^ .. .. .. .•..• .•. •....• ......I^ ...50^100^150^200^250Time (sec)300Figure 5.4.Plot of the velocity of the mass in figure 5.2 through time for the rigid lever(dotted line) and the compliant lever (solid line).continuing to rise when it reached 20 cm, while the rigid lever was slowingdown. This impression is supported in figure 5.4 which plots the velocity of themass during this modelled jump. At 180 ms the rigid lever had accelerated themass up to approximately 0.6 m/s. At 215 ms, however, the compliant leverhad accelerated the same mass to about 1.6 m/s - an improvement of 167%.Figure 5.4 does show how the peak velocity produced by the rigid lever wasabsolutely larger than that produced by the compliant lever (1.8 m/s vs. 1.6m/s), indicating that the energetic hysteresis of the rubber band produced alarger take-off velocity at the cost of lower peak velocity.Figure 5.5 is a plot of the acceleration of the mass during the modelledjump and provides an explanation of how this difference in performance isaccomplished. The differencing procedure is inherently noise producing assmall errors in measuring displacement are multiplied at each differencing step.Therefore, the data, represented by the dotted lines in figure 5.5, have beensmoothed with a three point moving average, represented by the solid lines.The mass connected to the rigid lever reached a very high acceleration atapproximately 60 ms and then proceeded to decline all the way to the take-offpoint of 180 ms. The mass connected to the compliant lever system did notreach peak acceleration until 150 ms and was positive all the way out to 215ms when the feet left the ground'.So the original hypothesis is supported. The built in compliance of thespindly tibiae may actually be acting as a time machine, delaying some of thetransfer of spring energy to a point later in the jump impulse where themechanical advantage is better able to use it. The mechanical advantage ofthe locust's tibiae has a larger range than my simple model, suggesting thatthey would get even greater benefit from this principle. However, for this model-161-50^100^150^200^250Time (sec)300Figure 5.5.Plot of the accelerations displayed by the mass in figure 5.2 through timefor the rigid lever and the compliant lever. Because serially differencing thedata in figures 5.3 & 5.4 produced a noisy signal, the original data (dotted lines)has been smoothed with a three-point moving average (solid lines).to work the benefits gained from delaying the decay of spring force mustoutweigh the hysteretic losses in loading and unloading the second spring. Wehave seen in chapter two that the losses in the tibiae from hysteresis are indeedvery low (ie. R-0.93). Thus, the legs' mechanical properties are well suited to thisstrategy. Therefore, I believe that the scaling that I observed in the locust legs'external dimensions may represent a design strategy that takes advantage ofthe deformability of long slender beams.Is there any evidence that this mechanism is actually used by the locust?Is there a characteristic signiture of the time machine that is demonstrated inthe data on jump performance? If the time delay in ground reaction force isgenerated by deforming the tibiae as bending springs, then we mighthypothesize that with increasing bending loads we would observe increasingtime delays in peak acceleration (Fig. 5.6a). To test this suggestion, I haveplotted the relative time within the period of the jump impulse where the peakin acceleration occurs as a function of the magnitude of the peak accelerationin figure 5.6b. The slope of the regression between the relative location of thepeak acceleration and the magnitude of acceleration has a slope of 4.97 x 10 -4(SE = 4.53 x 10 -5 , r2 = 0.482) and is significantly different from zero (t s = 10.96, df.= 129, p < 0.05). While the absolute value of the slope is low, it does mean thatover the range of accelerations produced by the locusts (45 - 230 m/s 2) theposition of the peak of acceleration moves from the 78% point in the impulsedurration to the 90% point in the impulse. Therefore, there is at least indirectevidence that the locusts' legs are acting as energy time machines, and it ispossible that they are obtaining the benefits in performance that are suggestedby the model.-163-Figure 5.6a.Diagramatic representation of the parameters reported in figure 5.6b. t totalis the total length of the impulse, and t peak is the length of time from thebegining of the impulse to the time of peak acceleration, and amm, is themagnitude of the peak acceleration.Figure 5.6b.The relationship between the normalized position in time of the peak ofacceleration produced in the jump (tpeak, ttotal) and the magnitude of the peak-acceleration (amax). The equation for the regression calculated for a sample offirst instars, fifth instars and adults Y = 0.784 + 4.967E-4 x X (F s = 120.090, df. = 1,129, r2 = 0.4821). This slope was significantly different from zero Os = 10.959, df.= 129, p < 0.05).ttotal10.5Time0^40^80^120^160^200^240Peak Acceleration (a.„ m/s 2).Degrees of freedom in design. Although this thesis has not concerneditself explicitly with the polymer physical chemistry or the composite materialmechanics of the cuticle, I would like to suggest that it is the use of this proteinexoskeleton that has allowed the distinctive scaling of skeletal morphology inthe locust. I believe that by introducing an additional degree of freedom in theadaptive design of skeletal structures (ie. functional alteration of materialproperties) the locust provides a window on a new approach to scaling.When an engineer designs structures for human use, the design will to a largeextent be determined by the building materials available: wood, steel, carbonfibre. The properties of these materials are relatively scale independent. Thefact that vertebrate skeletons are all built of the same material (ie. bone) hassuggested a set of models that allow only a discrete and finite set of designstrategies. By using a building material whose properties can be modulated,the locust has shown that a continuous set of strategies exist to solve theproblems for design imposed by scale effects. When human engineers alter thecomposition of composite materials to produce lighter and stronger structuresthan they could with steel, entirely new morphologies can be produced. To acertain extent this is an application of the principle that the locust has used inchoosing the morphological design program for the scaling of its legs.CHAPTER 6.CONCLUSIONSMy final findings are that the African desert locust has adopted a scalingprogram that is consistent with the principles of the elastic similarity modelproposed by McMahon (1973). However, this scaling is achieved withoutadopting a morphological scaling that is predicted by a traditional approachto elastic similarity. The scaling programme that the locust is expressingproduces relatively longer, more slender limb segments in the metathoracic andmesothoracic tibiae.I believe this scaling of external dimensions is accomplished by scaling thematerial stiffness of the cuticle material in addition to the dimensionsthemselves. This is a strategy allowed by the non-mineralized character of thecuticle and distinguishes the locust's scaling from vertebrate systems that otherresearchers have described. It has also been found that the exoskeletalcondition itself determines a separate strategy for being thin walled that is alsodistinct from the vertebrate, thick walled design.The reason for producing the relatively spindly scaling in the locust seemsto be related to the jumping mode of locomotion. 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